“The speaker continues to believe we need step-by-step efforts to fix our broken immigration system, but given the American people and Congress’s lack of trust that the president will enforce the law, it’s difficult to see how we make progress,” said Michael Steel, a spokesman for Mr. Boehner. As for the mockery, he added, “The speaker often says you only tease the ones you love.”

House Republicans, including Mr. Boehner, hoped to pass a series of narrow but related bills. But they also said that Mr. Obama’s repeated changes to the Affordable Care Act, as well as his State of the Union promise to use executive actions to circumvent Congress, have eroded trust and made any immigration deal with Republicans increasingly difficult.

Privately, Republicans also said that a broad immigration deal was unlikely in a midterm election year, and that the issue was too big and unwieldy to push through in the lame duck session after the November elections. Democrats pointed to a small window of opportunity — in June and July — when a deal could be reached, but they warned that it was rapidly closing, and that Republicans had not made any serious effort to work with them on a compromise.