WASHINGTON — Ten legislative days before funding would run out, Congress is heading toward another government shutdown showdown. Democrats and many Republicans are likely to refuse to go along with President Trump’s request for money for a border wall financed in part by outsize cuts to medical research. And the specter of another fight over Planned Parenthood funding is also in the offing.

Fresh off the humiliating implosion of the House health care bill last week, Mr. Trump appears to be courting another disaster. To help pay for his proposed border wall with Mexico, Mr. Trump has asked for $18 billion in cuts to domestic programs, including many with broad bipartisan support, and an additional $1.5 billion in funding as part of the spending bill to keep the government open for the rest of the year.

Among the cuts, the administration proposes a $1.2 billion reduction to the National Institutes of Health — which Congress enriched last year in a bill to fight cancer and other diseases — and a $2.8 billion reduction to the State Department and other international operations as well as major cuts in grants for transportation, infrastructure and housing.

Democrats said such a plan would arrive dead at the doorstep of the Senate, and Republicans on Tuesday sounded no more enthusiastic. “We just voted to plus up the N.I.H.,” said Senator John Cornyn, Republican of Texas, who has also been lukewarm on the border wall plan. “It would be difficult to get the votes to then cut it.”