Representative Andy Harris, a member of the Caucus, said: "I disregard and discount anything that involves the mainstream media press".
Asked in an interview with CBS if he held Mr Putin responsible for the interference, Mr Trump replied: "Well, I would, because he's in charge of the country".
He declined to answer whether Putin - who adamantly denies Moscow interfered with the USA election - is lying, however.
Speaking about his talks with Putin, Trump said: "As successful as North Atlantic Treaty Organisation was, I think this was our most successful visit".
He clarified his position the following day in a major u-turn by stating he did support the U.S. intelligence agencies after a public backlash.
The news comes after National Intelligence Director Dan Coats warned that cyberthreats from the Kremlin are "blinking red".
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"I thought the press conference was fine". I think we have excellent people in the agencies.
President Trump said that he told Russian leader Vladimir Putin during their summit in Helsinki that "we can't have meddling" in U.S. elections. Will a strong statement - you know - President Obama supposedly made a strong statement - nobody heard it.
"He made a disgusting statement, tried to back off, but couldn't even bring himself to back off", he told the Senate, adding that "Trump is afraid to confront Putin directly". And that statement was not acceptable.
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However, he did not respond when reporters asked him if he would condemn Mr Putin.
CHANG: Now, Trump also said today that he's been tougher on Russian Federation than any previous president. It's just too bad he couldn't have said all of this when Putin was standing next to him. I will tell you though, it better not be.
The president now says he accepts that meddling did happen, but insists it did not intefere with the result of the election which saw him become president. "But, no, I certainly don't have confidence in past people".
Sanders said "there was some conversation about" Putin's proposed exchange during Trump's summit with the Russian leader in Helsinki on Monday, where the two leaders spent about two hours together accompanied only by translators.
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On Twitter, Trump said he had received praise for his heavily criticized news conference with Putin. "And I think probably just concentrated on that the translation was being done effectively". But while the president has stopped short of saying whether he believes Putin did, in fact, direct Russian efforts, he did say that he holds Putin responsible. Sanders also said the White House believes the Russian threat of targeting USA elections "still exists".