Time to work more constructively with Russia - Donald Trump
But Mr Trump's position contrasts with some of his own senior officials.
The US ambassador to the UN, Nikki Haley said the US "can't trust Russia" and "won't ever trust Russia".
She told CNN that talking to Russia should never mean that the US "take our eyes off the ball".
And US Secretary of State Rex Tillerson said interference in the 2016 election remained an impediment to better relations with Russia.
A special prosecutor is investigating whether Trump associates colluded with alleged Russian efforts to influence last November's US election.
At the meeting at the G20 in Hamburg, both sides confirmed that the presidents had discussed Russia's alleged meddling in the US election, but at the time seemed unable to agree on the exact outcome.
Russian Foreign Minister Sergei Lavrov said Mr Trump accepted Mr Putin's assertions that the allegations were not true.
Mr Tillerson, meanwhile, said the two men held a "robust" discussion on the issue, and that Mr Trump had pressed the Russian leader on several occasions - but that an "intractable disagreement" might remain.
As part of his series of tweets on Sunday, Mr Trump said he "strongly pressed President Putin twice about Russian meddling on the issue" and that Mr Putin "vehemently denied it".
He said the pair discussed forming "an impenetrable Cyber Security unit", prompting derision on social media.