On Thursday's broadcast of "The Late Show", Tapper railed against the White House for barring CNN reporter Kaitlan Collins from an open event after she asked Trump about his former personal attorney Michael Cohen and Russian President Vladimir Putin.
But Trump's press conference, in which he suggested that he believed Putin's claims that Russian had no involvement in the USA election, sparked anger and frustration even among Republicans.
Trump still routinely casts doubt on the USA intelligence assessment that Russian Federation tried to sway the 2016 presidential election.
Trump has repeatedly denied his campaign colluded with Russian Federation and has characterized the investigation by Special Counsel Robert Mueller as a witch hunt.
Press Secretary Sarah Huckabee Sanders says in a statement that Trump "looks forward to having President Putin to Washington after the first of the year" and "is open to visiting Moscow upon receiving a formal invitation".
Trump has defended his conduct in a two-hour private meeting with Putin and at a joint news conference.
Mr. Putin said he understood Mr. Trump's desire to meet, but they need the "appropriate conditions" for another get-together.
"We are ready for such meetings. I am ready for this".
It's part of "a power game between Putin and Trump", said Dr. Alina Polyakova of the Brookings Institution.
The statement comes as Trump's apparent eagerness to embrace Putin is coming under increasing scrutiny after the summit between the two leaders last week in Helsinki. "Be my guest. He has such an invitation, I told him that", Mr. Putin said at a press conference in South Africa, where he is attending an economic conference.
Putin added: "I am ready to go to Washington".
"Russia isn't abandoning the dollar", Putin said in answer to a question about the sharp decline in its holdings of US Treasuries in April and May. Trump has since walked back his remarks. It led to him reading out a clarification in which he said he had misspoken, mixing up the word "would" with "wouldn't".
Moscow has portrayed its tensions with Washington as a result of Trump being hobbled by domestic political disputes and a widespread "Russophobia" perpetrated by holdovers from the Obama years - echoing Trump's penchant to blame his predecessor for many problems.
Mr Putin did not detail what "conditions" would have to be met.
He did not clarify if that was a face-to-face invitation to be followed by a formal one.