WATCH: Dan Coats's stunned reaction to being told Trump had invited Putin to the White House

WASHINGTON — Unbowed by swirling criticism of his summit encounter with Vladimir Putin, President Donald Trump swiftly invited the Russian leader to the White House this fall for a second get-together. Cleanup from the first has continued with no letup and Trump belatedly decided Putin's "incredible offer" of shared U.S.-Russia investigations was no good after all.

A White House meeting would be a dramatic extension of legitimacy to the Russian leader, who has long been isolated by the West for activities in Ukraine, Syria and beyond and is believed to have interfered in the 2016 presidential election that sent Trump to the presidency. No Russian leader has visited the White House in nearly a decade.

Trump asked National Security Adviser John Bolton to invite Putin, and "those discussions are already underway," Press Secretary Sarah Huckabee Sanders said Thursday. Trump earlier had tweeted that he looked forward to "our second meeting" as he defended his performance at Monday's summit, in which the two leaders conferred on a range of issues including terrorism, Israeli security, nuclear proliferation and North Korea.

"There are many answers, some easy and some hard, to these problems ... but they can ALL be solved!" Trump tweeted.

There was no immediate reaction from the Kremlin to the invitation.

News of the invite appeared to catch even the president's top intelligence official by surprise.

"Say that again," National Intelligence Director Dan Coats responded, when informed of the invitation during an appearance at the Aspen Security Forum in Colorado.

"OK," he continued, pausing for a deep breath. "That's going to be special."

The announcement came as the White House sought to clean up days of confounding post-summit Trump statements on Russian interference in the 2016 election. Trump's public doubting of Russia's responsibility in a joint news conference with Putin on Monday provoked withering criticism from Republicans as well as Democrats and forced the president to make a rare public admission of error.

Then on Thursday, the White House said Trump "disagrees" with Putin's offer to allow U.S. questioning of 12 Russians who have been indicted for election interference in exchange for Russian interviews with the former U.S. ambassador to Russia and other Americans the Kremlin accuses of unspecified crimes. Trump initially had described the idea as an "incredible offer."

The White House backtrack came just before the Senate voted overwhelmingly against the proposal. It was Congress' first formal rebuke of Trump's actions from the summit and its aftermath.

Asked about the Putin invitation, Alaska Republican Sen. Dan Sullivan said "I wouldn't do it, that's for damn sure."

"If the Russians want a better relationship, trips to the White House aren't going to help," he added. "They should stop invading their neighbors. They should stop meddling in our elections."