Since Monday's summit and joint news conference in Helsinki, Trump and his aides have been trying to control the damage done when he stunned the world by siding with Putin over US intelligence agencies on the question of Moscow's meddling in the 2016 USA election.
The White House announced Thursday that Putin has been invited to Washington this fall, a move that caught Trump's own director of national intelligence, Daniel Coats, by surprise.
Standing alongside Mr Putin, Mr Trump said he did not see any reason why it "would" be Russian Federation that meddled in the U.S. presidential election. The official, who spoke on condition of anonymity to discuss internal planning, said the talks would also cover nuclear proliferation, North Korea, Iran and Syria.
He said he still doesn't know what Trump discussed with Putin during the leaders' one-on-one meeting in Helsinki. No Russian leader has visited the White House in almost a decade.
Putin offered to let American officials watch Russian authorities question the 12 Russian military officers indicted in special counsel Robert Mueller's investigation.
The overall objective of Russia's election meddling in 2016 was to cause chaos and sow discord, said Homeland Security Secretary Kirstjen Nielsen at the Aspen Institute on Thursday.
"If we are going to start getting translator's notes, I think we are moving to a precedent that - unless some crime has been committed - is unprecedented and just not appropriate", Corker said on Thursday.
To Americans back home, particularly the opponents of Trump, the press conference gave them more fodder to criticize the United States president.
"The sentence should have been: 'I don't see any reason why it wouldn't be Russian Federation, '" he said.
Trump Hits ‘Fake News’ over North Korea Progress
US and South Korean officials say they are ready to help North Korea revive its economy if it gives up its nuclear program. The meeting was followed up by another working-level meeting on Monday to work out the next steps.
On Thursday, Trump said he was "not happy" that the Federal Reserve was raising interest rates, breaking with long-standing tradition at the White House of avoiding any influence on the US central bank.
He reiterated denials of Russian meddling in the 2016 presidential election - interrupting a questioner to say "We didn't interfere!" Trump's public doubting of Russia's responsibility in a joint news conference with Putin on Monday provoked withering criticism from Democrats and Republicans alike and forced the president to make a rare public admission of error. 42% of registered voters said they approved of Trump's overall job performance, according to a Reuters/Ipsos opinion poll taken after the summit.
"Now with that being said if it doesn't work out I'll be the worst enemy he's ever had".
Russia's Defense Ministry said it sent detailed proposals to Washington on organizing the return of more than 1.7 million refugees to Syria after agreements reached by Putin and Trump, RIA news agency said.
Antonov said it's important to "deal with the results" of their first summit before jumping too fast into a new one, but that "Russia was always open to such proposals".
"There were suggestions, comments, thoughts by President Putin with respect to that inquiry".
To be fair to Trump, he could not support a line of thought that he won the election with Russian support.