He also said Boris Johnson would make a “great prime minister”. But after talks with Mrs May he claimed “tremendous things” he said about her had been left out of the Sun story.
He said the paper’s story was “generally fine”, but “it didn’t put in what I said about the prime minister, and I said tremendous things”.
“Fortunately, we tend to record stories now so we have it for your enjoyment if you’d like it,” he said.
“We record when we deal with reporters, it’s called fake news and we solve a lot of problems with the good old recording instrument.”
The Sun’s write-up of its interview, which was recorded, quoted him describing her as a “nice person” and saying they get along “very nicely”.
Trump verbally sparred with the Sun reporter who had interviewed him, Tom Newton Dunn.
Mr Newton Dunn told the president that his positive remarks about Mrs May had been included in the story.
“If you reported them that’s good,” Mr Trump told the political correspondent.
“Thank you very much for saying that.”
He added: “I didn’t think they put it in, but that’s all right.
“They didn’t put it in the headline, I wish they put that in the headline, that’s one of those things.”
On Mrs May’s reaction, he added: “She’s a total professional because when I saw her this morning I said ‘I want to apologise because I said such good things about you’.
“She said: ‘Don’t worry it’s only the press.’ I thought that was very professional.”
During the press conference, Mr Trump – who regularly accuses the US media of “fake news” over its coverage of him – also clashed with a reporter from the American NBC network, describing it as “possibly worse than CNN”.
He also refused to take a question from CNN, saying: “CNN is fake news, I don’t take questions from fake news. Let’s take a question from a real network.”
He then took a question from Fox News regarding his upcoming meeting with Russian President Vladimir Putin.
Responding to Mr Trump’s comments about The Sun, a spokesman for the newspaper said: “We stand by our reporting and the quotes we used – including those where the president was positive about the prime minister, in both the paper and in our audio – and we’re delighted that the president essentially retracted his original charge against the paper later in the press conference.
“To say the president called us ‘fake news’ with any serious intent is, well… fake news.”