Mr Trump also told ITV's Good Morning Britain that Mrs May should seek a "carve out" in any Brexit deal with the European Union which would allow the United Kingdom to secure a trade agreement with the US.
"Whoever says we are foes is spreading fake news", Tusk, who is also a former Polish prime minister, wrote on Twitter.
The president flew in from Scotland, where he and his wife, Melania, spent the weekend at his Turnberry golf resort.
A second, expanded bilateral meeting with presidential aides followed the first session. "I'm not going in with high expectations".
But Trump crossed a line with his comments Monday, Murphy said, by "throwing the entire intelligence community under the bus". The U.S. ambassador to Russia, Jon Huntsman was also asked about Crimea Sunday. But as to impeachment, we'll see how this plays out.
The meeting comes against the backdrop of Trump feuding with European allies, 12 Russian military intelligence officers indicted stemming from the Mueller investigation into Russian meddling during the 2016 presidential election, and Democrats back home calling for Trump to cancel the meeting.
Trump and Putin will then have a larger meeting with other US and Russian officials, and end the day with a joint press conference. "I have president Putin, he just said it's not Russian Federation".
The first one he named was the European Union, traditionally a close ally to the U.S.
Critics and Trump's own advisers have urged Trump to use the summit to press Putin hard about "malign" activities.
This meeting is so informal that it's being described as a "getting to know you".
"It went well", he said. We are the two great nuclear powers, we have 90 percent of the world's nuclear [weapons].
Putin also addressed the issue of "so-called interference", saying he "had to repeat that the Russian state never interfered, and does not plan to interfere in internal American electoral process".
The grand jury charges shine a brighter spotlight on Trump's treatment of Putin, who has repeatedly denied Russian Federation sought to skew the election which Trump, a Republican, unexpectedly won.
"We will, of course, ask your favorite question about meddling", Trump told reporters last week.
"I hadn't thought of that", Trump replied. "I will be asking that question again".
Mr Putin arrived late for the meeting - another display of the Russian's leader famous lack of punctuality. He said in the CBS interview that he's going into the meeting with "low expectations" but believes "it's a good thing to meet".
Moscow has repeatedly slammed the deployment of USA missile systems in eastern Europe, while Washington has condemned Russia's alleged violations of the Intermediate-Range Nuclear Forces Treaty, under which the signatories are obliged to eliminate all short- and medium-range nuclear and conventional missiles.