“Elections have consequences,” Mick Mulvaney, the former House conservative who is now the head of the Office of Management and Budget, told The Associated Press in an interview. He offered Democrats a deal: The administration would support continued subsidies for millions of people receiving health insurance through the Affordable Care Act in exchange for Democrats agreeing to the initial wall funding.

Mr. Schumer called that proposal a nonstarter. Other Democrats suggested they saw no reason to accept that offer because Republicans are separately but simultaneously trying to unravel the health care program. They also believe that if White House refusal to fund the subsidies caused a collapse of the health insurance market, Republicans would get the political blame for that upheaval, as well.

Though a shutdown would most likely inflict significant political damage on Republicans, Democrats say they, too, want to keep the government running because shutdowns are costly, reckless and shake both public and international confidence in United States institutions.

But they also say that if the Trump administration and Republican leaders on Capitol Hill want to engage in a nasty political fight and shut the government down in a fit of pique over funding a border wall that even some Republicans do not back, then they are welcome to it. Both sides expect that if the impasse cannot be quickly breached, Congress will most likely extend current funding by a week to allow more time for a deal.

The question for Democrats is whether the White House is bluffing on its border wall demands so that the administration can later tell Republican voters that they pushed as hard as they could to get the money, particularly with so little to show legislatively for Mr. Trump’s first 100 days besides the confirmation of a Supreme Court justice, Neil M. Gorsuch. Mr. Trump is the wild card — would he shut down the government to prove he will not relent on his demands? And would Republican leaders stand with him?

Democrats have been on the sidelines while Republicans pushed through Trump nominees, rolled back federal regulations and sought unsuccessfully to disassemble the health care law without consultation with the minority party. But the appropriations process is a different animal.