Instead of answering the questions directly, Trump began by criticizing the Federal Bureau of Investigation for not confiscating the hacked e-mail server of the Democratic National Committee.
Putin has long insisted that the Kremlin had nothing to do with the hacking, denials that Trump said the Russian leader repeated during their private meeting Monday.
"I have real confidence in my intelligence people, but I must tell you that President Putin was extremely strong and powerful in his denial", Trump said.
"I get the feeling, I've seen it first-hand actually, sometimes the president cares more about how a leader treats him personally than forcefully getting out there and pushing against things that we know have harmed our nation", Corker said.
Richard Burr of North Carolina said his committee has no reason to doubt the intelligence community's "conclusion that President Putin ordered an influence campaign aimed at the 2016 elections".
Republican Rep. Justin Amash tweeted of the press conference that someone can be in favor of improved relations with Russian Federation and with Trump meeting Putin "and still think something is not right here". Trump said he held "both countries responsible" thinks the United States has been "fooling" and that "we're all to blame".
Putin also suggested that the two countries form a joint working group on cybersecurity that would look into the election issue, prompting Trump to say it was an "interesting idea".
5 takeaways from the Trump-Putin press conference
Asked whether Russian Federation was responsible at all, Trump said "we're all to blame" for the soured relations. This was after he tweeted earlier that Washington's "foolishness and stupidity" was the reason for bad relations.
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"I am asking Leader McConnell and Speaker Ryan. make sure these four things are done", he said. And what he did is an incredible offer.
The summit, which is being closely watched around the world, was not the first time Trump and Putin have held talks.
President Trump proved not only unable, but unwilling to stand up to Putin.
Despite his implication that Russian gas was a security risk for Germany and other recipients, Mr. Trump couched his response in terms of "competition" and Mr. Putin as a "competitor" of the US. "And it is why I believe we are at a critical point", Coats said.
"Because he talked about bringing the U.S. Russian Federation relationship back to normal", Putin said.
Former CIA director John Brennan, a Democrat and a frequent Trump critic, called Trump's performance at the news conference "nothing short of treasonous".
"I would guess [Putin] is having caviar right now", he said. He went on to say that Trump and Putin "seemed to be speaking from the same script" and Trump made a "conscious choice to defend a tyrant".
Some Republicans responded to Trump's remarks within moments of the press conference ending. "We've all been foolish", he said, before veering into discussion about his election victory.
Trump has been reluctant to blame Russian Federation for the hacks and again Monday refused to embrace the intelligence community assessment.