Congressional Republicans, already facing a difficult election landscape, confronted a prospect on Sunday they had worked feverishly to avoid: a threat by President Trump to shut down the government over funding for a border wall.

Last week, Republican leaders thought they had reached a deal with Mr. Trump to delay a confrontation on funding for the wall until after the November midterm elections, according to a person familiar with their discussion.

But Mr. Trump’s shutdown threat, in which he also demanded several pieces of a comprehensive immigration overhaul that is stalled in Congress, has opened the door to a politically bruising spending fight as the fiscal year ends in September.

With the election coming just weeks later, the party can ill afford a disruption that voters — already disgusted by Washington dysfunction — may hold the president accountable for.

A shutdown would also distract from Senate Republicans’ main business in September: their push to confirm Judge Brett M. Kavanaugh to the Supreme Court.

“We’re going to have a challenging midterm anyway, and I don’t see how putting the attention on shutting down the government when you control the government is going to help you,” Representative Tom Cole, Republican of Oklahoma, said in an interview.

Representative Steve Stivers of Ohio, the chairman of the committee charged with electing Republicans to the House, insisted that a shutdown was unlikely.

“I don’t think we’re going to shut down the government,” Mr. Stivers said on the ABC News program “This Week.” “You know, I think we’re going to make sure we keep the government open, but we’re going to get better policies on immigration.”

Democrats and Republicans have in fact made unusual progress on the 12 appropriations measures necessary to keep the government operating. Current funding for the government expires Sept. 30.

When asked if the inevitable battle between Republicans and Democrats over the wall would wait until after the midterm elections, Mr. McConnell said, “Probably, and that’s something we do have a disagreement on.”

Asked if he feared a government shutdown, Mr. McConnell was emphatic. “No, that’s not going to happen,” he said.

The House speaker, Paul D. Ryan of Wisconsin, said Mr. Trump was willing to wait for his wall funding. “As far as the wall is concerned, we’ve gotten some wall funding already underway,” he told reporters on Thursday before the House left for its August recess.

“I think it’s not a question of if, it’s a question of when,” he added. “And the president’s willing to be patient to make sure that we get what we need so that we can get that done, because border security’s extremely important.”

“The bill he’s describing only got 39 votes in the Senate floor,” said the spokesman, Matt House. “He should learn from his mistakes.”

“We walked the president through our strategy for appropriations before the fiscal year,” Mr. Ryan said. “He agreed with our strategy. So we think we have a unified strategy to make sure that we can get as many appropriation bills done as possible.”