The US president, who is currently in the UK on a four-day working visit, claimed that any attempt to keep close ties with the European Union would make a future trade deal with the US unlikely.
Mr Trump added that he had advised Ms May against pursuing a soft Brexit strategy but said she “didn’t listen” to him.
Ms May’s plans involve the UK seeking something very similar to a single market for goods and a “facilitated customs arrangement” which would essentially create a common rulebook with the EU.
Mr Trump said such a move would make any future negotiations with the US difficult.
“If they do a deal like that, we would be dealing with the European Union instead of dealing with the UK, so it will probably kill the deal,” he told The Sun.
Mr Trump also said he believed that Boris Johnson, the former foreign secretary, would make a “great prime minister”, describing him as a “very talented guy”.
“I have a lot of respect for Boris. He obviously likes me, and says very good things about me,” Mr Trump said.
“I was very saddened to see he was leaving government and I hope he goes back in at some point. I think he is a great representative for your country.”
In a wide-ranging interview the US president also took aim at Sadiq Khan, the mayor of London, who has long been a critic of Mr Trump.
He blamed the mayor for feeling “unwelcome” in the UK, in light of mass street protests and said he “used to love London”.
Mr Trump also added that he believed Mr Khan was responsible for the terror attacks that had taken place in the capital, saying that he had “done a very bad job on terrorism” as well as a “bad job on crime”.
“I think allowing millions and millions of people to come into Europe is very, very sad,” he told the paper.
“I look at cities in Europe, and I can be specific if you’d like. You have a mayor who has done a terrible job in London. He has done a terrible job.”
Mr Trump went further and claimed and that the UK and the rest of Europe was “losing its culture” because of immigration.
“Allowing the immigration to take place in Europe is a shame,” he said.
“I think it changed the fabric of Europe and, unless you act very quickly, it’s never going to be what it was and I don’t mean that in a positive way.
“So I think allowing millions and millions of people to come into Europe is very, very sad.
“I think you are losing your culture. Look around. You go through certain areas that didn’t exist ten or 15 years ago.”
On Thursday evening as the comments were made public, Mr Trump attended a lavish black tie welcome dinner at Blenheim Palace hosted by Ms May.
On Friday, Mr Trump will meet Ms May again to watch a special forces demonstration at a secret location in the English countryside, before they travel together to Chequers for “substantive bilateral talks” on foreign policy issues.
Following a joint press conference Mr Trump will meet the Queen at Windsor Castle.
He will then fly to Scotland, where he is expected to spend the weekend playing golf before departing on Sunday.
Mr Trump’s visit is set to be met by tens of thousands of protesters in central London, with a giant inflatable “Trump baby” set to be tethered near Parliament Square, depicting the president in a nappy.