“What I want is real money for the wall,” said Representative Jim Jordan of Ohio, a founder of the conservative House Freedom Caucus, who declined to say how much funding he would consider sufficient, but said $1.6 billion was not enough.

In a meeting in his office just off the Senate floor, Mr. Schumer flatly told Mr. Pence, Mr. Mulvaney and Mr. Kushner, who had requested to meet with him, that any measure that included money for the wall could not pass the Senate, and urged them to consider agreeing to one that omitted it but included funding for other forms of border security, according to a spokesman.

Complicating the chances of such a deal was the president’s own refusal to detail his bottom line in negotiations. During a meeting with Republican senators on Friday morning, Mr. Trump would not provide specifics about what kind of plan he could support, including how much money he would accept for fortifying the border, despite their repeated efforts to ascertain his conditions for a deal, according to a Senate official briefed on the session who insisted on anonymity to describe it. The president talked at length about the wall and repeatedly pressed the senators about eliminating the filibuster so they could fund it with 51 votes.

Senator John Cornyn of Texas, the chamber’s No. 2 Republican, appeared practically giddy at the prospect that Mr. Trump’s aides were engaging in serious talks with lawmakers, saying, “The fact that that’s happening represents real progress,” and adding that he was “so happy about it.”

Without a strategy for averting a shutdown, Mr. Trump spent the day maneuvering to ensure that Democrats would shoulder the blame, notwithstanding his public courting of the dysfunctional denouement.

He began the day warning on Twitter that a partial government shutdown “will last for a very long time.”