On Wednesday, he sparked fresh controversy by appearing to say "no" when asked if Russian Federation was still targeting the United States, again contradicting intelligence agencies. The White House says Trump is weighing the offer.
Trump and Putin both praised US-Russia military ties and spoke vaguely about Syria to reporters on Monday after their talks in Helsinki, Finland, noting shared concerns for the security of Syria's neighbor, Israel.
Yesterday, Mr Trump accused opponents of preferring to go to war rather than seeing good relations with Russian Federation.
Of greatest interest to Western observers will likely be videos showing Mr Putin's nuclear-powered cruise missile - now being called Burevestnik, which has meanings ranging from "storm-bringer" and "thunderbird" to "petrel" - and the Avangard ground-launched hypersonic missile. Trump made a rare personal correction on Tuesday, telling reporters he misspoke and meant to say "wouldn't" instead of saying he didn't see any reason why Russian Federation "would" interfere. They are pushing so recklessly hard and hate the fact that I'll probably have a good relationship with Putin. Unlike many other news organizations, we have not put up a paywall.
Mr Trump has faced fierce criticism for contradicting his own intelligence agencies by refusing to blame Russian Federation for meddling in the 2016 election.
KIM: He's happy, but I think he's also quite cautious.
European Commission President Jean-Claude Juncker is scheduled to meet with Mr Trump at the White House next Wednesday to discuss trade and other issues. But he said Trump "is not making the problem better, he's making it worse".
Few details about what was discussed or agreed upon by Trump and Putin at a one-on-one session at the summit, with only translators present, have emerged.
Putin said he would permit the 12 to be questioned inside Russia if the United States allowed Russia to question former US envoy to Russia Michael McFaul and 11 others in Moscow's case against billionaire investor and human rights activist William Browder, the driving force behind Magnitsky Act sanctions on Russian officials passed by the US Congress.
That rhetoric marked a turnabout from Trump's first, upbeat description of the sit-down. When asked by a reporter Tuesday if Russian Federation was continuing to target the U.S., Trump replied, "No".
Press Secretary Sarah Huckabee Sanders: "I had a chance to speak with the president after his comments, and the president was-said, "Thank you very much, ' and was saying 'no" to answering questions".
The zigzagging laid bare the White House's search for a path out of trouble that has dogged the administration's discussions of Russian Federation from the start, but spiraled after Trump's trip to Helsinki. Jeff Glor: "But you haven't condemned Putin specifically".
Meanwhile, Russian politicians are angry at proposals by USA lawmakers to question Trump's translator about what the men discussed privately before their news conference. Instead, he went into a rambling response, including demands for investigation of Hillary Clinton's email server and his description of Putin's "extremely strong and powerful" denial of meddling.
That's a change from Trump's initial upbeat description of his meeting with Russia's president.