“If he wants a solution, that’s a step backward,” said Senator John Cornyn of Texas, the No. 2 Republican in the Senate.
Further muddying the conversation, the White House refuses to acknowledge the offer that Senate leaders on both sides of the aisle have confirmed.
“Senator Schumer is trying to rescind an offer that he never made in the first place, and misled the public about,” said Sarah Huckabee Sanders, the White House press secretary.
Roughly 800,000 young people brought to this country illegally have been protected from deportation under an Obama-era initiative, Deferred Action for Childhood Arrivals, or DACA. But Mr. Trump rescinded the program in September, giving Congress six months — until March 5 — to come up with a replacement.
Now the Senate has a new immigration deadline: Feb. 8, the date that a stopgap spending measure approved on Monday will expire. Mr. McConnell has said that if the fate of the young undocumented immigrants is not resolved by then, he will allow the floor debate on immigration.
“What came out of this mess was a date certain to move on immigration,” said Senator Lindsey Graham, Republican of South Carolina, a longtime advocate of an immigration overhaul who has been deeply involved in the bipartisan talks. “That’s positive. Whether you’re a Democrat or a Republican, we’re now moving on immigration. That wouldn’t have happened without this conflict.”
But hard-liners on immigration were not budging.
“I think the top priority for Senate Democrats is amnesty for people here illegally,” said Senator Ted Cruz, Republican of Texas. “They just shut the government down over amnesty, and so I don’t think Senator Schumer and Senate Democrats are interested in securing the border.”
Such voices would dominate debate in the House, where twice, in the Obama and George W. Bush presidencies, bipartisan comprehensive overhauls of the nation’s immigration laws have died. But senators, given the green light by their leaders, pressed ahead.
If nothing else, passage of a bipartisan immigration bill in the Senate would shift the political focus to the House, where the bulk of the 2018 election will be fought.
As lawmakers are racing to a conclusion, so are the courts. After a federal judge ruled that Mr. Trump could not summarily end the DACA program, the Justice Department appealed to the Supreme Court. The Supreme Court said Tuesday that it would consider the case on an expedited basis, and ordered both sides to submit briefs by Feb. 2.
One outstanding question is what role Mr. Trump will play in any DACA negotiations; the president has been unclear about what he wants, and senators have said that until he makes his wishes known, it will be difficult to reach an agreement.
But one thing is certain: The president and his allies on Capitol Hill will almost certainly reject any proposal that does not include funding for the “big, beautiful wall” that Mr. Trump promised to build along the southern border, though he had said Mexico would pay for it.
Late Tuesday night, Mr. Trump reiterated his request for tougher border security. He posted on Twitter that “Cryin’ Chuck Schumer fully understands, especially after his humiliating defeat, that if there is no Wall, there is no DACA.”
The administration has asked Congress for $18 billion to build the wall; Mr. Cornyn said that in negotiations with the White House before the shutdown, Mr. Schumer had offered the president $25 billion. A spokesman for Mr. Schumer declined to comment but did not dispute the figure.
But Mr. Schumer said he rescinded the offer because Mr. Trump had rejected the rest of the immigration package.
“The wall offer was made as part of a broader deal. The president rejected that broader deal, so the offer is off the table,” Mr. Schumer said.
Mr. Graham said he had a message for the administration. “To my friends at the White House: You’ve been all over the board,” he said. “You haven’t been a reliable partner, and the Senate is going to move. Please be constructive as we go forward. If you’ve got any ideas, let us know, but the Senate is going to lead on this issue.”
The week before the shutdown, a bipartisan group of six senators led by Richard J. Durbin, Democrat of Illinois, and Mr. Graham proposed a measure that would have provided a path to citizenship for DACA recipients, as well as more than $2.5 billion for border security and a grant of protected status for the parents of the Dreamers, who would be barred from sponsoring their parents for citizenship.
But Mr. Trump objected to that plan during a White House meeting in which he used vulgar, insulting language to describe African immigrants. That rejection and the ensuing firestorm set back the talks.
At the White House on Tuesday, Ms. Sanders underscored the administration’s opposition to the proposal. From the briefing room podium, Ms. Sanders rejected it as a potential part of a framework for a new deal.
“It’s totally unacceptable to the president and should be declared dead on arrival,” she told reporters.
But in a reflection of how much the debate has revolved around bluster instead of substance, Ms. Sanders said at the same briefing that she did not think the sides were “that far apart.”
That was not the sense on Capitol Hill on Tuesday. Lawmakers who have been working to resolve the DACA recipients’ status confessed that they did not know where they were headed. Asked what the next step would be, Mr. Durbin shrugged.
“I don’t know the answer,” he said. “I went to a meeting yesterday, a bipartisan meeting of senators, and we’ll see where it goes.”
While Mr. Durbin has suggested that the Durbin-Graham proposal should be the starting point for negotiations, Senator Jeff Flake, an Arizona Republican who has been a key participant in the bipartisan talks and has a history of bucking his party on immigration, said that would not be the case.
“We’ve negotiated a lot of these things,” Mr. Flake said. “We know where the contours are. But we’re not starting with any template.”
Talks that helped end the shutdown were led by a group of about 25 senators — Republicans, Democrats and an independent — who called themselves the Common Sense Coalition. They met in the office of Senator Susan Collins, Republican of Maine.
Ms. Collins said Tuesday that while she did not intend for that group to pre-empt other efforts on immigration, its members “will be getting together to figure out if there’s a role for us to play.”
But any effort to pull the Senate to the center on immigration will almost certainly be met with resistance from hard-liners, among them Senator Tom Cotton, Republican of Arkansas, who said he was “already in the center of the immigration conversation.”