The Republican National Committee had high hopes that Donald J. Trump would deliver a compassionate and measured speech about immigration on Wednesday, and prepared to lavish praise on the candidate on the party’s Twitter account.

So when Mr. Trump instead offered a fiery denunciation of migrant criminals and suggested deporting Hillary Clinton, Reince Priebus, the party chairman, signaled that aides should scrap the plan, and the committee made no statement at all.

The evening tore a painful new wound in Mr. Trump’s relationship with the Republican National Committee, imperiling his most important remaining political alliance.

Mr. Priebus and his organization have been steadfastly supportive of Mr. Trump, defending him in public and spending millions of dollars to aid him. But the collaboration between Mr. Trump’s campaign and Mr. Priebus’s committee has grown strained over the last month, according to six senior Republicans with detailed knowledge of both groups, some of whom asked to speak anonymously for fear of exacerbating tensions.