The White House rejected on Friday a Vladimir Putin-backed effort to hold a referendum in eastern Ukraine on the region's future, distancing itself from the idea in the aftermath of President Donald Trump's controversial summit with the Russian leader.

Russia's ambassador to the US, Anatoly Antonov, said the two leaders had discussed the possibility of a referendum in separatist-leaning eastern Ukraine during their Helsinki summit.

But Trump's National Security Council spokesman Garrett Marquis said agreements between Russia and the Ukrainian government for resolving the conflict in the Donbas region do not include any such option and any effort to organise a "so-called referendum" would have "no legitimacy."

The back-and-forth came as the White House outlined the agenda for a proposed second summit between Trump and Putin - in Washington this fall - that would focus on national security. Moscow signaled its openness to a second formal meeting between the two leaders as criticism of Trump over his first major session with his Russian counterpart kept up in the US

Trump left the White House for his New Jersey golf club for the weekend. Once he got there, he returned to Twitter to complain about news coverage of Monday's meeting.

"I got severely criticised by the Fake News Media for being too nice to President Putin," he tweeted. "In the Old Days they would call it Diplomacy. If I was loud & vicious, I would have been criticised for being too tough."

A White House official said the next Trump-Putin meeting would address national security concerns they discussed in Helsinki, including Russian meddling. The official did not specify if that meant Russia's interference in US elections. The official, who spoke on condition of anonymity to discuss planning, said the talks would also cover nuclear proliferation, North Korea, Iran and Syria.

One stop Putin almost surely won't make is Capitol Hill.

House Democratic leader Nancy Pelosi urged House Speaker Paul Ryan to make clear that Putin wouldn't be invited to address Congress if he visits Washington.

She said Trump's "frightened fawning over Putin is an embarrassment and a grave threat to our democracy."

Secretary of State Mike Pompeo had a sunnier view of the likely second get-together.

He said at the United Nations he was "happy that the two leaders of two very important countries are continuing to meet. If that meeting takes place in Washington, I think it is all to the good. Those conversations are incredibly important."

It was not clear whether such a meeting would take place before or after the November congressional elections in the US

A White House meeting would be a dramatic extension of legitimacy to the Russian leader, who has long been isolated by the West for activities in Ukraine, Syria and beyond and is believed to have interfered in the 2016 presidential election that sent Trump to the presidency. No Russian leader has visited the White House in nearly a decade.

US officials have been quiet on what, if anything, the two leaders agreed to in Helsinki during a two-hour, one-on-one meeting. Director of National Intelligence Dan Coats said Thursday he had yet to be briefed on the private session.