In the interview, the president posed the idea that Mr. Farage, the leader of the newly founded Brexit Party who has been a battering ram to traditional conservative politics in Britain and a rival and irritant to Mrs. May, should take over the negotiations.

Mr. Trump, who once suggested that Mr. Farage become ambassador to the United States, praised him as a “very smart person” who could help guide Britain after its decision to leave the European Union. Mr. Farage has backed such a departure for decades.

“I like Nigel a lot. He has a lot to offer,” Mr. Trump said. “He is a very smart person. They won’t bring him in. Think how well they would do if they did. They just haven’t figured that out yet.”