But the House’s approach, to combine short-term funding to keep the government open with long-term funding for the military, was long ago rejected by most Senate Democrats, who want to pair an increase in military spending with a similar increase in domestic spending. The measure would need 60 votes to pass the Senate, where Republicans hold only 51 seats.
Still, the inclusion of the military funding is expected to secure the votes of reluctant Republicans in the House, including members of the conservative Freedom Caucus, allowing party leaders to push the measure through their chamber even if the vast majority of Democrats oppose it.
“It’s a good play call,” said Representative Mark Meadows, Republican of North Carolina and the chairman of the Freedom Caucus, who just last week had suggested his caucus was not likely to support another temporary spending measure.
On Monday, Senator Chuck Schumer of New York, the Democratic leader, warned Speaker Paul D. Ryan of Wisconsin against such an approach, saying it amounted to “barreling headfirst into a dead end.”
“If he lets the Freedom Caucus be the tail that wags the dog, there’s no way we’ll reach an agreement that can pass the Senate,” Mr. Schumer said on the Senate floor.
Lacking the votes to approve the temporary spending measure paired with long-term military funding, the Senate could strip out the military portion and send the measure back to the House.
“I think everyone understands that this will probably end up being a Ping-Pong situation, and we’ll see where the ball lands,” said Representative Carlos Curbelo, Republican of Florida.
But Mr. Meadows was eager for a fight over the military funding. “That’s going to be up to Democrats to decide whether they want to continue to keep our military men and women hostage,” he said.
To bring an end to last month’s brief shutdown, lawmakers approved a temporary spending bill that keeps the government open through the end of Thursday. Now another stopgap measure is needed. Adding to the tight timeline facing lawmakers, House Democrats are scheduled to hold a retreat in Maryland beginning Wednesday.
“Here we are again,” Senator John Cornyn of Texas, the No. 2 Senate Republican, said on the Senate floor on Monday, before complaining about the use of one stopgap bill after another. “Governing is not a merry-go-round,” he said.
The stopgap bills have been needed because congressional leaders have yet to reach a deal to raise the caps on military and domestic spending, an impasse that has kept them from negotiating a long-term spending bill. There were signs of hope on Monday that an agreement could be within reach.
Once a deal on those spending levels has been reached, lawmakers can put together a long-term funding bill that would stretch for the rest of the fiscal year. In the near future, lawmakers also need to raise the debt limit.
Another big question mark is the issue of immigration, which has helped make a deal on the spending caps elusive and played a central role in last month’s shutdown. Lawmakers are looking to take action following Mr. Trump’s decision last year to end an Obama-era program, Deferred Action for Childhood Arrivals, or DACA, that shields from deportation young immigrants who were brought to the country illegally as children.
On Monday, Senator John McCain, the Arizona Republican who has been absent from the Senate while he battles brain cancer, teamed up with Senator Chris Coons, Democrat of Delaware, to offer a narrow immigration bill that would protect the young undocumented immigrants, known as Dreamers, from deportation, while also seeking to strengthen border security.
Senator Mitch McConnell of Kentucky, the majority leader, has promised an open debate on immigration legislation in the Senate if lawmakers do not reach a deal on that subject by Thursday. Mr. Coons told reporters that his bill could serve as the underlying measure to form the foundation of that debate, calling it a “strong starting point.”
But the McCain-Coons bill, intended as the Senate counterpart to a bipartisan bill in the House, is slimmer than the framework proposed by Mr. Trump. Notably, it does not include the $25 billion Mr. Trump requested for border security, including for a wall along the southern border with Mexico.
In addition, the measure does not address the diversity visa lottery, which fosters immigration from countries that are underrepresented and which Mr. Trump wants to abolish, or family-based migration, which would be severely limited under the White House proposal.
On Monday, the White House rejected the McCain-Coons plan, just as it turned away a different bipartisan proposal put forth last month by a group of senators led by Lindsey Graham, Republican of South Carolina, and Richard J. Durbin, Democrat of Illinois.
Mr. Trump, without mentioning Mr. McCain or Mr. Coons by name, wrote on Twitter on Monday that any DACA deal that does not include a border wall was “a total waste of time.”