“I think it’s not a very desirable outcome,” he said.

But Senator Chuck Schumer of New York, the Democratic leader, said his party would be open to the idea. “We’d certainly very seriously look at it,” he told reporters.

After days of eerie quiet with a partial shutdown looming, the burst of activity suggested that all sides might be preparing for a brief holiday reprieve from their stalemate.

Ms. Sanders offered the first glimmers of a way out of the impasse in an interview Tuesday morning on Fox News, in which she said Mr. Trump — who only a week ago said he would be proud to force a shutdown over wall funding — did not want to see government funding lapse. She said the president was open to spending options short of the $5 billion lump sum he has demanded, and would find “different funding sources” to finance the wall.

“The president has asked every agency to look and see if they have money that could be used for that purpose,” she told reporters later.

Still, that would require approval from Congress, which Democrats said they would not grant.

In an evening tweet, the president appeared to continue to inch closer to the restrictions of the $1.6 billion funding in the Senate bill, claiming that “we are not building a Concrete Wall.” (Mr. Trump has mentioned concrete in his frequent, but fluctuating, descriptions of a wall.)

“We are building artistically designed steel slats, so that you can easily see through it,” the president wrote. “It will be beautiful and, at the same time, give our Country the security that our citizens deserve.”

Mr. McConnell had suggested a plan that would have provided $1.6 billion for border security but prohibited spending it on a wall, while allowing Mr. Trump to spend as much as $1 billion in unspent funds from other agencies on his immigration priorities. Mr. Schumer and Representative Nancy Pelosi of California, the presumptive incoming speaker, quickly rejected it.