To the surprise of many, the White House was reported to be reviewing the demand.
Putin gave his latest account of the meeting during at a conference with top Russian ambassadors and officials at the Foreign Ministry in Moscow, the people said, asking not to be identified discussing the president's comments to the part of the session that was closed to the public.
Trump clapped back at his critics and said he was "far tougher on Russia" than his predecessors.
How did Trump respond to Putin's suggestion?
That stands in sharp contrast to the rain of criticism the US president has faced at home.
In the White House on Tuesday, Trump attracted more criticism for saying that it "could be other people also" who interfered with USA elections.
In one Twitter post, Trump blamed the media. He blames Trump's opponents.
"The ball's in your court", said Putin with an air of forced joviality as he presented Trump with a gift to mark their meeting.
Putin indicated he would permit American law enforcement officials to observe questioning by Russian officials of the indicted Russians and vice versa for Russian investigations.
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President Donald Trump distanced himself from allegations that he was cozying up to Russian Federation.
In Moscow, Putin said "powerful" USA forces were trying to sabotage the summit's achievements, but said he and Trump had begun to improve US-Russia ties anyway. "So certainly as the leader of the country you would have to hold him responsible".
While he and Trump seem to be cut from the same cloth, they come from very different worlds. Russian officials worry that domestic turmoil in the US will hamper potential future cooperation on Syria or arms control discussed at the summit.
President Trump is being increasingly vocal about his annoyance at this resistance to his diplomatic overtures, despite avoiding a war with Russian Federation seemingly being an obvious part of long-term U.S. strategic interests.
Mueller is investigating possible collusion between Trump's campaign and Russian Federation.
"He's made things much worse", Gabuev said.
Trump believes this amounts to pushing "recklessly hard" for war, saying "they hate the fact that I'll probably have a good relationship with Putin". At the time, White House officials worked to convince a skeptical president that the Nordic capital would serve as a more effective backdrop - and warned of a firestorm should a West Wing meeting go through. John Brennan, director of the Central Intelligence Agency under Barack Obama, even said Trump's words constituted "treason".
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US intelligence officials have said Russian election interference efforts are continuing and now target the upcoming congressional elections in November. No Russian leader has visited the White House in almost a decade.
Putin also insisted that Russian Federation remains open to further contacts with the USA on an equal and mutually beneficial basis.
At the summit with Trump in Helsinki, Putin proposed letting Russians observe interrogations of former U.S. Ambassador to Russia Michael McFaul and other Americans.
The Time magazine cover was released just hours after the first portion of the interview aired on Wednesday, and both fueled the discussion about Mr. Trump's remarks during the press conference with Mr. Putin.
Over the course of my career as an undercover officer in the C.I.A., I saw Russian intelligence manipulate many people.
GREENE: He sounds pretty happy with the summit, the Russian president does.
President Putin reacted aggressively to questions from USA reporters from the new agencies Reuters and the Associated Press. Standing alongside Putin, Trump called the idea "an incredible offer".
"I don't know what happened in that meeting", he said.
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