The compromise bill, meanwhile, is facing trouble with conservatives, who would be taking a political risk by supporting a bill that has been derided as offering “amnesty” to young undocumented immigrants. Republican leaders appear to be hoping that with a bit more time, the legislation can be modified to win over reluctant members, with the addition of an E-Verify provision as one possible change.

Republicans in the Senate are backing narrow legislation to ensure that children are not taken from their parents at the border. But they were in disagreement with Democrats over how to do it, and senators from both parties tried Thursday to tamp down expectations that the chamber would pass a solution that would make it into law quickly.

In the House, Republican leaders were trying to keep the compromise bill alive after it had appeared headed toward defeat.

“I am a big fat no — capital letters,” Representative Lou Barletta, Republican of Pennsylvania, said before the vote was delayed until next week. “I’m going to encourage other people to vote no because it doesn’t stop amnesty.”

Early in the day, Mr. Trump did not appear to help matters, venting his frustration on Twitter that even if the House passed immigration legislation, it would require Democratic support to clear the Senate. In effect, the president suggested that voting for the bill might be a pointless exercise — not exactly a persuasive message for conservatives on the fence about whether to support a measure that could rile up their party’s right flank.