Congressional leaders have ruled out allowing a vote on the bipartisan Gang of Six immigration deal, backing President Trump but leaving Congress with no clear pathway to a deal on illegal immigrant Dreamers and putting the government closer to a partial shutdown.

With a Friday deadline looming, House Republicans said they would try to pass a bill to keep the government open four more weeks, without taking any action on immigration.

After caving a month ago, Democratic leaders insisted this time that they would not accept another stopgap spending bill without at least the outlines of an immigration deal. Now, that vow is being put to the test.

Both sides still said they were hopeful that something could be worked out, though they were far apart on what an immigration deal would look like.

“There is a deal to be had this week,” said Senate Minority Leader Charles E. Schumer, New York Democrat.

He and fellow Democrats said there’s only one option: Republicans accept an immigration plan worked out by the Gang of Six senators, three Republicans and three Democrats, that offers a generous pathway to citizenship for as many as 2 million illegal immigrants, combined with a small part of Mr. Trump’s border wall and tweaks to other immigration policy.

But Republican leaders said that deal didn’t give Mr. Trump enough concessions and wouldn’t win over enough Republicans to pass it.

“I think the meeting at the White House on Thursday demonstrated that it is going to take more than six people to agree to a deal for it to pass the House and the Senate,” said Senate Majority Whip John Cornyn, Texas Republican. “We welcome their ideas, but it is going to have to be part of a larger negotiation.”

Mr. Cornyn said neither Senate Majority Leader Mitch McConnell, Kentucky Republican, nor House Speaker Paul D. Ryan, Wisconsin Republican, would put the Gang of Six deal on the floor for votes.

Sens. Lindsey Graham, South Carolina Republican, and Richard J. Durbin, Illinois Democrat, said they would persevere anyway as leaders of the Gang of Six.

They are counting on most Democrats to back their deal, given the generous amnesty, but they need to persuade Republicans to buy in.

“To my Republican friends, I say: We’re not going to get the sun, moon and the stars. We should push for the best deal we can get, but we shouldn’t let the perfect be the enemy of the good, so let’s be realistic,” Mr. Graham said.

He and Mr. Durbin thought they had earned Mr. Trump’s support for their bill last week.

They said they alerted the White House to their deal and rushed over for a meeting — but the president had soured by the time they arrived. They blamed Mr. Trump’s aides, including Chief of Staff John F. Kelly and senior adviser Stephen Miller, for poisoning the talks.

“By the time we got there, something had happened,” said Mr. Graham, confronting Homeland Security Secretary Kirstjen Neilsen at a Judiciary Committee hearing.

Mr. Graham said the negotiations have “turned into an S-show,” and he pleaded with the president to re-engage.

Mr. Schumer said the president will forgo his demands and sign on to the Graham-Durbin bill if he wants people to stop calling him a bigot.

“This may be the last train leaving the station. President Trump needs to get on board,” he said.

But it’s unclear whether Mr. Schumer has leverage to force a deal.

While Democrats insist there is an end-of-week deadline for action on immigration and it must be tied to the spending bill, Republicans say the issues are completely different.

They are pushing for a continuing resolution — dubbed a “CR” in Capitol-speak — to keep the government open beyond the Friday deadline when funding runs out.

“I am skeptical that you’re going to be able to do much on immigration and resolve it as part of [a] CR,” said Rep. Ryan A. Costello, Pennsylvania Republican.

Republican leaders have said they have until March 5 — the final day of Mr. Trump’s phaseout of the DACA deportation amnesty that is protecting 690,000 Dreamers.

They might have even longer than that. A federal judge last week ruled the phaseout illegal and ordered Ms. Nielsen to again begin accepting renewal applications under Deferred Action for Childhood Arrivals. The Justice Department appealed that ruling directly to the Supreme Court on Tuesday.

Instead of immigration, Republicans were hoping to avoid a shutdown and entice support for a temporary spending bill by adding a six-year extension of the popular Children’s Health Insurance Program and a delay of unpopular Obamacare taxes into the package.

They already have one Democratic vote in hand.

“I will not vote to shut down the government,” said Sen. Joe Manchin III, West Virginia Democrat. “I can’t go home and explain that. It doesn’t make sense.”

Sen. Doug Jones, a Democrat who last month narrowly won an upset victory in a special election in solidly conservative Alabama, dodged questions about how his constituents would respond to a government shutdown over Dreamers.

“I’m not going to talk about hypotheticals,” he said. “We were down in Alabama and had a wonderful weekend [with] MLK Day.”

Sen. James M. Inhofe, Oklahoma Republican, said he remained hopeful that lawmakers would come together to avoid a shutdown.

“Nobody really wants it on either side. It’s a stubborn game,” he said.

• Stephen Dinan contributed to this article.

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