Updated at 2:05 p.m.
The White House closed the door on Thursday to making a former American diplomat or other Americans available to Russian interrogation following a bilateral request from Russian President Vladimir Putin.
Press secretary Sarah Sanders said in an announcement on Thursday afternoon that President Trump now "disagrees" with Putin's proposal from their summit in Helsinki on Monday that the two governments exchange people of interest to each other's investigators.
"It is a proposal that was made in sincerity by President Putin, but President Trump disagrees with it," Sanders said. "Hopefully President Putin will have the 12 identified Russians come to the United States to prove their innocence or guilt."
Sanders alluded to 12 intelligence officers in Russia's military spy agency, the GRU, who were charged in an indictment on Friday that linked them to the cyberattacks in 2016 against political targets and state-level elections targets.
The Senate was set to vote on a measure condemning the earlier idea that the White House had held open — that Russians might get to interrogate Americans if American investigators were able to talk with the Russian suspects — and individual lawmakers and others also had spoken out.
A former U.S. Ambassador to Russia, Michael McFaul, says Putin is obsessed with an imaginary scheme to undermine the Russian government that McFaul supposedly conducted from the U.S. Embassy in Moscow, and McFaul worried about Trump permitting this whole story to take flight.
Granting any credibility to the scheme means the White House is "playing into Russia's hands," Senate Foreign Relations Committee Chairman Bob Corker, R-Tenn., told reporters on Thursday.
Corker said that although the president has wide discretion to conduct foreign affairs, he might have gone too far in Helsinki and that he wants to question Secretary of State Mike Pompeo about the summit at a hearing next week.
"The president does have the ability to meet with anybody he wishes," Corker said. "I think this one, especially the presentation that was made at the end, was disconcerting to most Americans and certainly was to me," Corker said.
"We're going to have Pompeo come in next week, and I think we'll be much better informed when that is over, as to what the intentions are, where they're trying to take this relationship, was there anything that was agreed to privately in those meetings."
Sen. Marco Rubio, R-Fla., wrote on Twitter that he thought the White House must "publicly and unequivocally rule it out."
And former Secretary of State Hillary Clinton also weighed in on McFaul's behalf, calling him a "patriot ... To see the White House even hesitate to defend a diplomat is deeply troubling."
The prospect that the United States might grant access to McFaul, in exchange for American investigators talking with Russians linked to cyberattacks, is viewed as unlikely.
But Trump called it an "incredible offer" on Monday in his press conference with Putin in Helsinki, and Sanders seemed to suggest Wednesday that the administration was evaluating it.
Critics want the White House to close the door completely and make clear there is no scenario in which a former American official could get caught up in some kind of bilateral exchange of this kind.
One issue for administration officials, however, appears to be how receptive Trump has remained to the idea.
"The president is going to meet with his team, and we'll let you know when we have an announcement on that," Sanders told reporters Wednesday.
Another issue is that this idea is so novel that so long as it hasn't been officially ruled out from the highest level, no one seems to be sure how it might practically work.
State Department spokeswoman Heather Nauert said Wednesday that although Putin's charges against McFaul and other Americans are "absolutely absurd," she was not prepared to speak on behalf of the department and say that her agency opposed the idea of some kind of handover.
"I believe some of this would fall under the Department of Justice, so I'd have to loop in the Department of Justice on this," she said. "This is something that just came out."
NPR correspondents Deirdre Walsh and Miles Parks contributed to this report.