Trump broached that topic first, according to Tillerson, and repeatedly pressed Putin on the matter, even as the Russian president denied involvement.
Public images of the interaction between the brash property tycoon and ice-cool ex-KGB agent are likely to be dissected frame by frame for any sign of rapprochement or estrangement.
Putin told reporters Saturday that Trump appeared to agree with Moscow's position that it did not interfere in last year's election.
Leaders of the Group of 20 (G20) except US President Donald Trump have reaffirmed their commitment to the Paris Climate Change Agreement, saying it was "irreversible" in a declaration at the end of the summit in Hamburg, Germany.
The two sides directed their governments to work out details of the agreement not to use cyberattacks as a tool of political subversion.
Obama after the election punished Russia for its meddling by imposing economic sanctions on the country and seizing two Russian compounds in the U.S. that intelligence officials say were used for spying. Trump did, however, win a concession in the final summit communique on the need to promote what he calls "fair" trade.
The approach towards Syria seems to be more straight forward from the Trump administration, said Nicholas Heras, a Fellow at the Center for a New American Security (CNAS).
Trump's long list of meetings with world leaders on Saturday also included Japanese Prime Minister Shinzo Abe and Prime Minister Lee Hsien Loong of Singapore.
A conversation between the leaders of Russia and the United States was allotted about an hour. "As I think he has in the past", Tillerson said.
Swalwell responded to a question on whether he believes Putin's or Trump's version of events by saying, "I think there's an opportunity now". If you've been paying attention, you've heard it all before - in fact four times before.
"We assess Russian President Vladimir Putin ordered an influence campaign in 2016 aimed at the U.S. presidential election". Both the U.S. and Russia oppose the Islamic State group in Syria.
The video has been rapidly shared on social media, with many Twitter users finding humour in the situation. How do we live with one another?
Defending series champion Shane van Gisbergen gave Triple Eight two drivers on the podium with his third-placed finish.
"Neither one of them wanted to stop", he said. "But the nature of policymaking is such that leaders are judged by present costs and not by those that could occur down the line".
Trump also met Mexican counterpart Enrique Pena Nieto for the first time as president and insisted that Mexico would "absolutely" pay for his planned border wall. "We as Europeans have to take our fate into our own hands", she said.
He warned that clinching more agreements between Russia and the US will be a rather tricky task.
Earlier, Mr Trump said he had a "tremendous meeting" with Mr Putin on Friday, which included the Syrian ceasefire.
"It can only be a good thing if there is an honest, frank dialogue", she said. "If Kim wants to be North Korea's developmental dictator, the United States' best long-term strategy is to help him do so".
The president said that he expected a trade deal that would bolster Britain after its exit from the EU could be agreed to "very, very quickly".