“The president is the chief executive who is going to be a part of solving this problem,” said Senator Thom Tillis, Republican of North Carolina, “so I must trust him.”

But hours later, the president contradicted Mr. Pence, saying in McAllen, Tex., that he would be open to a “broader” immigration deal that would “simultaneously” deal with the Dreamers and a wall — if senators would bring him one they could agree on.

Again, Republicans were left baffled by a president who has pitched himself as an expert in the art of the deal.

“I wouldn’t want to comment on his unique style of negotiation and communications,” Mr. Rooney said. “Sometimes it’s worked very effectively for him, and sometimes it has confused a lot of people.”

The concern goes beyond Mr. Trump and Mr. Pence. Jared Kushner, his son-in-law and senior adviser, has also been asked to help resolve the wall impasse, but Mr. Kushner has no experience in crisis negotiations on Capitol Hill, and his attempts at intervention have borne little fruit. The day before Mr. Trump’s prime-time Oval Office address to the nation, Mr. Kushner called Senator Joe Manchin III of West Virginia, a centrist known for his enthusiasm for bipartisan deals, and said the president was firmly committed to his position on the wall and did not plan to budge, according to two people familiar with the conversation, who spoke on the condition of anonymity because the call was private.

Mr. Kushner has little relationship with Mr. Manchin, but he left the senator with the impression that the White House believed public opinion would be on the president’s side after the speech, and that Democrats would simply have to relent. That did not go over well.