During a conversation aboard Air Force One flying into Paris, Trump said he was willing to engage the Russian leader despite controversy over the country's involvement in the 2016 US election.
Asked whether he would invite Putin, Trump said he would "at the right time. I don't think this is the right time, but the answer is yes, I would."
Trump argued that shunning Putin would be the easy thing to do, but it would not be smart, according to the comments released by the White House.
Trump has been embroiled in a scandal over his campaign's ties with Russia -- which US intelligence agencies say tried to tip the 2016 US election in his favour.
Trump denies allegations of collusion, but the recent publication of his son's emails show Donald Jr did solicit dirt on Hillary Clinton from a Russian contact.
"Look, it's very easy for me to say absolutely, I won't. That's the easy thing for me to do, but that's the stupid thing to do," Trump said.
"Let's be the smart people not the stupid people. The easiest thing for me to tell you is that I would never invite him. We will never ever talk to Russia. That all of my friends in Congress will say, oh he's so wonderful, he's so wonderful.
"If you don't have dialogue, you have to be fools. Fools."
During a meeting with French President Emmanuel Macron on Thursday, Trump said that a ceasefire in Syria -- which he agreed with Putin last week -- was evidence that engagement works.
He added that a second ceasefire in another region of Syria was also under consideration.
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