In his letter, Mr. Johnson said the House would instead move ahead next week with its drive to impeach Alejandro N. Mayorkas, the homeland security secretary, and doubled down on his demands that Congress embrace either an immigration crackdown bill the House passed last year or an equally severe measure.
“Since the day I became speaker, I have assured our Senate colleagues the House would not accept any counterproposal if it would not actually solve the problems that have been created by this administration’s subversive policies,” he wrote.
The letter reflected a stance Mr. Johnson and other hard-right Republicans in the House have maintained for months, repeatedly dismissing the border enforcement measures under discussion in the Senate as insufficient. It came as Republican proponents of the deal in the Senate toiled to build needed G.O.P. support to push it forward. That task has grown much more difficult as Mr. Trump, who has savaged the plan, has gained ground in his quest for the party’s presidential nomination.
Senator Mitch McConnell of Kentucky, the minority leader, told fellow Republicans behind closed doors this week that Mr. Trump’s hostility to the plan and his growing dominance in the primary had put them “in a quandary.”
Mr. McConnell, a chief Republican proponent of sending more aid to Ukraine, has been a vocal supporter of the border deal that members of his party have insisted upon as the price of their backing for continued assistance for Kyiv.