US President Donald Trump has asked his national security adviser, John Bolton, to invite Russian President Vladimir Putin to Washington later in the year, the White House says.

Key points: US National Intelligence director expresses surprise at invitation

US National Intelligence director expresses surprise at invitation Trump rejects Putin offer on interviewing Americans

Trump rejects Putin offer on interviewing Americans US President has faced backlash in US over comments following Helsinki summit

"President Trump asked @Ambjohnbolton to invite President Putin to Washington in the fall and those discussions are already underway," White House spokeswoman Sarah Sanders said in a tweet.

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News of the invite appeared to catch even the President's top intelligence official by surprise.

"Say that again," National Intelligence director Dan Coats responded when a moderator broke into their conversation at a security forum to inform him of the invitation.

"OK," he continued, pausing for a deep breath.

"That's going to be special."

Mr Trump had earlier tweeted that he looked forward to "our second meeting" as he defended his performance at Monday's summit.

The top Senate Democrat Chuck Schumer said Mr Trump should not meet privately with Mr Putin until Americans learn what happened at their first summit in Helsinki earlier this week.

"Until we know what happened at that two-hour meeting in Helsinki, the President should have no more one-on-one interactions with Putin. In the United States, in Russia, or anywhere else," Senator Schumer said in a statement.

Even Mr Coats made the surprise admission that he did not know what was said in one-on-one talks between the two leaders in Helsinki.

"Well, you're right, I don't know what happened at that meeting," Mr Coats said in response to a question at the security forum.

Sorry, this video has expired Donald Trump backtracks on comments about Russia's election meddling

The coveted invitation to Washington was sure to be seen as a victory by Mr Putin, whose last official visit to the United States was in July 2007, when he spent two days at the Bush family compound in Kennebunkport, Maine.

Mr Trump and Mr Putin both described their meeting in Helsinki as a success.

But Mr Trump has faced a backlash in the US for a number of comments he made following their talks, including his apparent backing of Mr Putin over the assessments of US intelligence agencies on Russian meddling in the 2016 US presidential election.

Mr Putin accused forces in the US of trying to undermine the success of the summit, but said the two leaders had begun to improve US-Russia ties anyway.

In his first public comments about the meeting, Mr Putin told Russian diplomats that US-Russian relations are "in some ways worse than during the Cold War", but that the talks had allowed them to start on "the path to positive change".

But he said "powerful" US forces were trying to sabotage what the summit had achieved.

Mr Putin did not name names, but spoke of US politicians who put their "narrow party interests" above the best interests of the US and were powerful enough to be able to foist their questionable "stories" on millions of Americans.

Trump rejects Putin offer on interviewing Americans

The White House says Donald Trump has rejected a proposal from Vladimir Putin on questioning American citizens. ( AP/Pablo Martinez Monsivais )

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In the latest fallout from the Helsinki summit, Mr Trump rejected Mr Putin's proposal that Russian authorities be allowed to question American citizens, the White House said, after the offer drew fierce criticism in the US.

The proposal was to allow the US to question 12 Russians accused of interfering in the 2016 election in exchange for permitting Russia to interview Americans the Kremlin accuses of unspecified crimes.

Mr Trump on Monday called the idea "an incredible offer" and on Wednesday the White House said the President was considering the proposal, drawing a barrage of criticism from his fellow Republicans as well as Democrats.

But White House spokeswoman Sarah Sanders said the offer was now off the table.

"It is a proposal that was made in sincerity by President Putin, but President Trump disagrees with it," she said in a statement.

"Hopefully President Putin will have the 12 identified Russians come to the United States to prove their innocence or guilt."

Mr Putin had described the offer as one that would permit American law enforcement officials to observe questioning by Russian officials of 12 Russian intelligence officers indicted last Friday in the United States on charges of meddling in the 2016 US presidential election in exchange for letting Russian investigators question Americans on other matters.

He specifically mentioned a case involving London-based financier Bill Browder, a one-time investor in Russia.

Sorry, this video has expired Mr Putin's "number one enemy" Bill Browder slams Trump meeting.

Reuters/AP