Putin unveiled the proposal in a joint press conference with Trump on Monday following their summit meeting in Helsinki, Finland.
It comes after Mr Putin, asked in Helsinki whether he would extradite the 12 Russian intelligence agents indicted in the United States for hacking Democratic Party computers, said he would meet the U.S. government "halfway".
Amid the unrelenting criticism of Trump's performance inside the United States, several lawmakers wanted the interpreter of Trump in his one-on-one meeting with Putin to testify before Congress for what exactly the two leaders had said in Helsinki. Trump tweeted. The announcement of the invitation came as the White House sought to clean up days of confounding statements on Russian interference in the 2016 election that sent Trump to the presidency.
Mr Putin said Russian Federation wanted to question U.S. nationals the Kremlin accuses of "illegal actions".
White House officials didn't immediately respond to a request for comment.
The 12 Russians is a reference to those indicted last week by Mueller for their role in interfering in the 2016 US election. "In the United States, in Russian Federation, or anywhere else", Schumer said in a statement.
Konstantin Kosachev, head of the upper house of parliament's foreign affairs committee, said the idea sets a unsafe precedent that threats the "the whole idea of diplomacy", according to Russian news agencies. But I really want to see the server.
"Certain forces are trying to disavow the results of the meeting in Helsinki", and hamper progress on what they discussed, such as limiting their nuclear arsenals or ending the war in Syria, Putin said.
He said he hadn't seen Trump's invitation himself, but that "Russia was always open to such proposals". According to one source, the meeting was kept private because Trump asked Putin to give him some time to consider the offer. The U.S. and Russia have been on opposing sides of the conflict in Ukraine, unleashed after a popular uprising against a pro-Russian president and Russia's annexation of Crimea in 2014.
The scale of the bipartisan outcry at Trump's stance toward Putin has only been rivaled by his 2017 waffling over condemning white supremacist demonstrators in Charlottesville, Virginia.
Onstage at the Aspen Security Forum, Andrea Mitchell of NBC News interrupted her discussion with Coats to say: "We have some breaking news". "That's going to be special". "The American people should rest assured", Pompeo said.
Asked by Glor again on Wednesday whether he thinks the threat is ongoing, Mr. Trump said he would accept Coats' conclusion. Russian officials worry that domestic turmoil in the US will hamper potential future cooperation on Syria or arms control discussed at the summit.
The idea of allowing a foreign power to quiz US citizens sparked outrage and the US Senate voted 98-0 against it. Secretary of State Mike Pompeo said it was "not going to happen".