The pair met for talks lasting more than two hours last week during the G20 summit in Hamburg.
Mr Trump claimed he questioned his Russian counterpart for the first 20 or 25 minutes of the meeting, asking Mr Putin if he was involved in what US intelligence believe was meddling in the presidential campaign.
"I said, did you do it?," Mr Trump said in an interview with the Reuters news agency.
"And he said no, I did not. Absolutely not. I then asked him a second time in a totally different way. He said absolutely not."
Asked if he believed the Mr Putin’s denial, Mr Trump said: “Something happened and we have to find out what it is because we can't allow a thing like that to happen to our election process."
Mr Trump also spoke about his son Donald Trump Jr's meeting last year with a Russian lawyer, saying he had been unaware of it until just a few days ago.
“No, that I didn’t know until a couple of days ago when I heard about this,” he said.
Mr Trump also said he did not fault his son for holding the meeting.
"I think many people would have held that meeting," the President said.
US intelligence agencies and members of the Trump team have accused the Kremlin and its affiliates of interfering in the election, but Mr Trump himself has wavered on the subject, at times suggesting that others might have been involved.
Asked whether he felt he could trust Putin, Mr Trump replied: "I am not a person who goes around trusting lots of people. But he’s the leader of Russia. It is the second most powerful nuclear power on earth.
“I am the leader of the United States. I love my country. He loves his country.”
Mr Trump again insisted there was no collusion between his campaign and Russia.
"There was zero coordination,” he said. “It’s the dumbest thing I’ve ever heard.”
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