Republicans were similarly upbeat. “I’m optimistic that very soon we’ll be able to reach an agreement,” said Senator Mitch McConnell of Kentucky, the majority leader.

But if the bipartisan deal falls through, lawmakers have no clear plan to keep the government open past Thursday, with the parties disagreeing over spending priorities and the president lobbing verbal bombs from the White House.

“I would shut it down over this issue,” Mr. Trump said as he demanded an immigration deal on his terms. “If we don’t straighten out our border, we don’t have a country.”

Mr. Trump’s call for a shutdown was not the first time he had brandished the threat of closing down the government. He mused on Twitter last year that the country “needs a good ‘shutdown,’” a suggestion Democrats did not forget.

And on Tuesday, in a striking moment, a lawmaker from the president’s own party, Representative Barbara Comstock of Virginia, pushed back at the White House meeting.

“We don’t need a government shutdown on this,” said Ms. Comstock, a top Democratic target in the midterm elections. She represents a moderate district in Northern Virginia, an area that is home to many federal workers.

Soon after, Mr. Trump interrupted her. “You can say what you want,” he said. “We’re not getting support from the Democrats on this legislation.”

It has been less than three weeks since the last shutdown — a three-day closing that ended after Democrats won a promise from Mr. McConnell to have the Senate consider legislation to protect young immigrants who were brought to the United States illegally as children.

Mr. McConnell pledged that the Senate would turn to immigration if no deal had been reached on that subject by Thursday, when the current government funding measure is set to expire. This time, Senate Democrats have shown no appetite to force a shutdown.

But the fate of the young immigrants, known as Dreamers, remains highly uncertain, especially given Mr. Trump’s insistence that Democrats agree to build a wall on the Mexican border and enact other tough immigration policies.

The White House chief of staff, John F. Kelly, also took a hard line on immigration during a visit to the Capitol on Tuesday.

He said Mr. Trump was unlikely to extend a March 5 deadline, when the Obama-era program to protect Dreamers — Deferred Action for Childhood Arrivals, or DACA — is set to expire. And he said the president had been generous in his offer to give 1.8 million young undocumented immigrants a path to citizenship in exchange for a series of hard-line immigration policy changes.

“There are 690,000 official DACA registrants, and the president sent over what amounts to be two and a half times that number, to 1.8 million,” Mr. Kelly told reporters. “The difference between 690 and 1.8 million were the people that some would say were too afraid to sign up, others would say were too lazy to get off their asses, but they didn’t sign up.”

Neither party seemed to have any idea how the immigration debate would play out in the Senate in the days to come. Mr. McConnell has promised a free and open debate, with senators allowed to offer amendments to whatever measure is brought to the floor. But what the starting bill would look like remained a mystery.

“That’s the $64,000 question,” said Senator John Cornyn of Texas, the No. 2 Republican in the Senate. “Everybody wants to know, and Senator McConnell hasn’t told us.”

Mr. McConnell gave little hint of how the debate would proceed.

“In the Senate, on those rare occasions when we have these kind of open debates, whoever gets to 60 wins,” he told reporters. “And it’ll be an opportunity for 1,000 flowers to bloom.”

But with less than 72 hours remaining to avert a shutdown, congressional Republicans were moving on an entirely different track from the president, and immigration was not part of the equation as the House moved ahead with the stopgap spending bill.

The House voted 245 to 182 to approve the bill, with most Democrats voting against it.

House Republicans were hoping to pressure Senate Democrats to go along or face harsh political consequences. The bill would keep the government open through March 23.

“Quite literally the safety of our service members and the security of our country is at stake,” Speaker Paul D. Ryan of Wisconsin said.

But 44 members of the Senate Democratic caucus signed a letter in December opposing the House Republican approach. Passing the bill in the Senate will require 60 votes, meaning that those 44 Democrats, if they stick together, could block it.

“This is not a serious bill,” said Representative Nita M. Lowey of New York, the top Democrat on the House Appropriations Committee. “It is nothing more than a political ploy that will place us on the brink of another shutdown.”

If a deal on the spending caps is reached, lawmakers could approve legislation that includes that agreement along with a temporary measure to keep the government open. The package could also end up carrying other items as well, including disaster aid in response to last year’s hurricanes and perhaps an increase to the statutory limit on the government’s borrowing authority.

Mr. Trump’s effusive praise of his chief of staff came during a visit to Pennsylvania where he was appearing to promote his tax cut and boost Rick Saccone, a Republican state legislator running for Congress in a special election in March.

Asked whether he minded Mr. Kelly telling lawmakers he had not been fully informed about immigration, Mr. Trump said: “No, he did not say that. He didn’t say it the way you would like him to say it.”

But according to one person familiar with the president’s thinking, Mr. Trump was livid when he learned that Mr. Kelly had described him as “evolving” in his immigration position. Throughout the evening on Wednesday, Mr. Trump fielded calls from allies who described Mr. Kelly’s comments to Congress as undermining the president, stoking Mr. Trump’s fury.

The president — who never likes it when someone characterizes his thinking — vented his anger to Mr. Kelly and to allies, according to the person familiar with the president’s thinking. It was similar to a moment during the campaign in April of 2016, when Paul Manafort — who had just been hired to the Trump campaign — was caught on tape at a meeting with Republican National Committee members saying of Mr. Trump, “the part he’s been playing is evolving.”

When a president’s chief of staff speaks to members of Congress, it should be a “consistent message,” Representative Henry Cuellar, a Democrat of Texas, said on Thursday in an interview with CNN. Mr. Cuellar, who attended Wednesday’s meeting with Mr. Kelly, said the inconsistency “makes it hard” to negotiate.

Lawmakers who attended the meeting on Wednesday described Mr. Kelly’s remarks. Representative Luis V. Gutiérrez, a Democrat of Illinois who was at the meeting, said Mr. Kelly told the group that “a 50-foot wall from sea to shining sea isn’t what we’re going to build.”

Mr. Gutiérrez told reporters that Mr. Kelly referenced Mr. Trump’s campaign promises to build a wall and said, “There were statements made about the wall that were not informed statements.”

In an interview with Fox News on Wednesday night, Mr. Kelly defended his comments to the Democratic lawmakers and said, “There’s been an evolutionary process that this president has gone through, as a campaign, and I pointed out to all of the members that were in the room that they all say things during the course of campaigns that may or may not be fully informed.”

Mr. Kelly also said the president has “evolved in the way he’s looked at things,” in reference to what should constitute a wall along the southern border.

Congress is currently working on a deal that would protect some 780,000 young immigrants from deportation. During the meeting on Wednesday, Mr. Kelly relayed confidence that negotiations would move forward for a permanent solution to preserve protections under the Deferred Action for Childhood Arrivals program, or DACA, the Obama-era policy that Mr. Trump moved in September to end.

In a Twitter post later on Thursday morning, Mr. Trump said there would be no such deal without a wall.

Donald J. Trump

@realDonaldTrump

We need the Wall for the safety and security of our country. We need the Wall to help stop the massive inflow of drugs from Mexico, now rated the number one most dangerous country in the world. If there is no Wall, there is no Deal!

5:16 AM – Jan 18, 2018

The president had recently started to water down his statements about building a wall and told lawmakers last week that 2,000 miles of wall would not be needed because of natural barriers. The wall is estimated to cost $18 billion over the next 10 years and cover 900 miles of the southern border, according to a spending plan submitted to Congress earlier this year.

The White House director of legislative affairs, Marc T. Short, pushed back against the Democrats’ accounts of Mr. Kelly’s remarks on Wednesday. “I don’t recall General Kelly saying the president was uninformed,” Mr. Short told CNN in an interview.

Accounts of what was said and by whom during recent high-level meetings on immigration policies have already hurt prospects for a broad spending and immigration deal to be achieved by Friday. Some participants in an Oval Office meeting last week said Mr. Trump called African nations “shitholes” in a discussion about immigration, reigniting concerns about the president’s racially tinged language about immigrants. Others said the president used the term “shithouse.”

The absence of an agreement could lead to a government shutdown, a risk that Mr. Trump may have increased later on Thursday morning when he wrote another Twitter post that blew up Republican plans to keep the government running.

Democrats have said they will not support a government spending plan that does not address the fate of the young immigrants, known as Dreamers, who could face deportation as soon as March.

Mr. Trump’s morning Twitter posts and his anger with Mr. Kelly about his chief of staff’s remarks to lawmakers come after weeks of the president fielding complaints from allies and staff members about Mr. Kelly’s restrictive influence. Mr. Trump has heard repeatedly that Mr. Kelly is trying to isolate him, and that senior aides to the president are frustrated that they cannot speak directly to Mr. Trump and must go through a filter.

People have warned the president that he faces a massive staff exodus among senior officials, in part as a result of the working environment that Mr. Kelly has created.