But the president's national security adviser, John Bolton, said that a Washington-based meeting between Trump and Putin wouldn't happen until 2019 due to special counsel Robert Mueller's "witch hunt".
Trump and Putin may meet on the sidelines of an worldwide conference at some point, the Russian government had said, but the upshot of Friday's back-and-forth was that both leaders want another bilateral summit as well at some point.
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.
Mr Trump drew ire at the meeting when he contradicted USA intelligence agencies by backing away from blaming Russian Federation for meddling in the 2016 election.
"He has a desire to have further meetings, I am ready for that".
Putin told reporters on Friday: "We need for the appropriate conditions to exist, to be created, including in our countries" for more U.S. -Russia summits.
Putin went on to say Russian Federation was ready to greet Trump in Moscow. "We are ready for such meetings. Be my guest. He has such an invitation, I told him that".
Referring to the proposal for a meeting in the United States, Putin said: "I am ready to go to Washington".
The White House statement came after National Security Adviser John Bolton said earlier this week that a meeting with Putin in the USA would be pushed off until next year - after the conclusion of the Federal Bureau of Investigation probe into Russian interference in the 2016 election. "I repeat once again, if the right conditions for work are created".
You can see in pictures of Putin and Trump that the ball bore a logo for a near-field communication tag.
Putin, who has ruled Russian Federation for most of the last two decades, also lauded Mr. Trump for what he said was a unique quality in a politician; keeping his promises.
He said he wanted to talk about the controversial Iran nuclear deal, saying: "It's a subject for debate and negotiations".
"You can criticize him for what he is doing, and some people do criticize him, however, one thing remains absolutely clear - he is committed to fulfilling his campaign promises".
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