US President Donald Trump wants to work "constructively" with Russia, including on cybersecurity, despite confronting his Russian counterpart Vladimir Putin over alleged meddling in last year's American elections.

While ruling out easing sanctions so long as the two countries remain at odds over Syria and Ukraine, Trump said on Sunday it was time for US-Russia relations to move forward, even though members of his own party said he should be mulling over new punishments.

Two days after his first face-to-face talks with his Russian counterpart on the sidelines of the G20 summit in Germany, Trump said he confronted Putin 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 in Hamburg. "He vehemently denied it. I've already given my opinion."

But the US president said the two sides could work together on some areas, including 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."

...We negotiated a ceasefire in parts of Syria which will save lives. Now it is time to move forward in working constructively with Russia! — Donald J. Trump (@realDonaldTrump) July 9, 2017

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".

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 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.

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"There is no reason to consider that Russia interfered in the electoral process in the US," Putin said in comments carried on Saturday by the Interfax news agency. "We did not interfere."

When asked later by reporters about Putin's description of the talks with Trump, White House officials did not refute the assertion.

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.

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 fuelled 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.