The interview continued, but grew heated after the special counsel’s investigation of Mr Trump came up.
“You can go to prison instead,” the president said.
The interview took place in the White House Oval Office on Monday, two days before Mr Trump travelled to Florida to formally launch his 2020 re-election campaign.
Trump asked to speak off-the-record, including in the moments after showing the Time reporters a letter he said was from the leader of North Korea.
“Here’s a letter, OK, now I’m going to show you this letter,” said the Republican president.
“So this was written by Kim Jong-un. It was delivered to me yesterday. By hand.”
As the conversation turned to his potential Democratic challengers in next year’s election, White House press secretary Sarah Sanders told the journalists: “You can’t take a picture of that, sorry.”
“What was that?!” Mr Trump exclaims.
The interview continues as the reporters ask about who Mr Trump sees as his toughest political opponent for 2020.
“Uh, I don’t know. Look, I think I’ve done so much. Could you bring the list of things, please, give me four of them,” Mr Trump tells an aide, who quickly presents one, saying it was made “for today”.
The president becomes more prickly when Time challenges an assertion by him.
The reporter reminds Mr Trump that some of his aides had “testified under oath, under threat of prison time” that the president had tried to influence the investigation by special counsel Robert Mueller.
Mr Trump snaps back: “Excuse me… Well, you can go to prison instead, because if you use, if you use the photograph you took of the letter that I gave you confidentially.
“I didn’t give it to you to take photographs of it – So don’t play that game with me.”
The journalist from Time responds: “I’m sorry, Mr President. Were you threatening me with prison time?”
Mr Trump replies: “Well, I told you the following. I told you you can look at this off the record. That doesn’t mean you take out your camera and start taking pictures of it. OK?
“So I hope you don’t have a picture of it.
“I know you were very quick to pull it out – even you were surprised to see that.
“You can’t do that stuff. So go have fun with your story. Because I’m sure it will be the 28th horrible story I have in Time magazine.”
Mr Trump has long has a fascination with Time magazine, and has had fake issues printed with himself on the cover and framed at his golf resorts.
He was named person of the year in 2016.
The next year, he claimed to have turned down an offer to be the magazine’s person of the year again, which Time’s editorial staff said was untrue.
In 2018, Time’s annual person of the year award was dedicated to persecuted journalists.