While ruling out easing sanctions so long as the two countries remain at odds over Syria and Ukraine, Trump said it was time for US-Russia relations to move forward, even though members of his own party said he should be mulling new punishments.
Two days after his first face-to-face talks with his Russian counterpart, Trump said he confronted Putin when they met in Germany over evidence from the intelligence agencies that Moscow meddled in the US elections.
"I strongly pressed President Putin twice about Russian meddling in our election," he said of Friday's meeting on the sidelines of the G20 summit in Hamburg. "He vehemently denied it. I've already given my opinion..."
But after saying that he had called out Putin over the election, Trump said they could work together on some areas, including on Syria, where he said a ceasefire which began on Sunday would "save lives."
"Now it is time to move forward in working constructively with Russia!"
In a series of early-morning tweets on his return from Europe, Trump said he and Putin had talked about the idea of setting up what he called "an impenetrable cyber security unit" to prevent hacking in future elections.
But senior Republican senators, including former presidential candidate John McCain, poured scorn on the idea.
Lindsey Graham, a member of the Senate's armed services committee, said on NBC that the cyber idea was "not the dumbest idea I've ever heard, but it's pretty close."
Voice dripping with sarcasm, McCain told a CBS interviewer that he was "sure that Vladimir Putin could be of enormous assistance in that effort, since he's doing the hacking."
The US and Russian sides have issued sharply conflicting accounts of Friday's meeting, with Putin saying on Saturday that Trump had been "satisfied" by his denials of any Russian interference in the polls.
The US president has previously equivocated over whether Russia did try to tilt the outcome of last November's election contest against Hillary Clinton in his favor, amid an investigation into whether members of Trump's campaign team actively colluded with Moscow.
So his public assessment that Russia did meddle has triggered calls to bring in more sanctions.
"So far they have not paid a single price for that," McCain said.
Moscow has warned that a program of existing sanctions, which were mainly imposed over Russia's annexation of Crimea in Ukraine, threatens their whole relationship.
Asked on Sunday whether new sanctions were in the pipeline, US Treasury Secretary Steve Mnuchin told ABC television: "We have sanctions that are already on the table and we expect to enforce those sanctions."
Mnuchin also insisted that Russia and the US could work together on cyber security.
"What we want to make sure is that we coordinate with Russia, that we're focused on cyber security together, that we make sure that they never interfere in any democratic elections," he said.
"This is like any other strategic alliance, whether we're doing military exercises with our allies or anything else. This is about having capabilities to make sure we both fight cyber (crime) together, which I think is a very significant accomplishment for President Trump."
Syria has been a particular source of friction between the two countries, as Russia is a close ally of President Bashar al-Assad.
Moscow was furious when the Trump administration launched a cruise missile strike against Syrian forces in April, in retaliation for what Washington said was a chemical weapons attack by Assad's regime against civilians.
While saying sanctions were not discussed at the meeting with Putin, Trump indicated that Moscow could not expect any relief "until the Ukrainian & Syrian problems are solved."
But while the brokering of a Syria ceasefire by the two countries -- along with Jordan -- means that one area of friction could be eased, the differences over Ukraine remain stark.
Speaking on a visit to Ukraine, US Secretary of State Rex Tillerson urged Russia to take action to ease the bloody separatist conflict in the country's east, which Kiev and the West believe is being fueled by Moscow.
"It is necessary for Russia to take the first step to de-escalate the situation in eastern Ukraine," Tillerson said as he made his first visit as Washington's top diplomat to Kiev.
Meanwhile, investigators on Sunday were probing possible collusion between the Trump campaign and Russia according to The New York Times.
The daily reported that son Donald Trump Jr., son-in-law Jared Kushner and then-Trump campaign chairman Paul Manafort met in New York in June 2016 with a Russian lawyer with Kremlin connections, the earliest such contact yet reported.
The younger Trump said in a statement that the meeting was to discuss the adoption of Russian children by American couples under a program Putin had suspended. Trump said he had been asked to attend, "and there was no follow-up."
(Except for the headline, this story has not been edited by NDTV staff and is published from a syndicated feed.)