White House press secretary Sarah Huckabee Sanders said they had not received a formal invitation. Trump has said he wants to improve ties between the two nuclear powers, whose relations have dipped to a post-Cold War low in recent years.
"We are ready to invite President Trump to Moscow", Mr Putin said during a press conference in Johannesburg today. He also said that it was in the interests of Russian Federation and the USA to extend arms control treaties.
Speaking to reporters at the Pentagon, Mr Mattis said: "I am considering meeting with my counterpart, but there's been no decision".
He was referring to the federal investigation into Russia's interference in the 2016 election and whether Trump's campaign assisted or was aware of the efforts.
CBS News' Steven Portnoy, who was serving as the radio network pool reporter for the day, said Collins "was no less willing to leave" the office than "any other member of the larger-than-usual press contingent". It led to him reading out a clarification in which he said he had misspoken, mixing up the word "would" with "wouldn't".
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Moscow's military support starting in 2015 of Syrian President Bashar al-Assad in the Syrian war, Russia's 2014 annexation of Crimea from Ukraine and the conclusions of US intelligence agencies that Russian Federation meddled in the election have soured relations.
"(Trump) has a desire to conduct meetings in the future and I am ready", Putin said, extending the invitation and reiterating his desire to go to Washington.
On Wednesday, National Security Adviser John Bolton said that the president instead "believes that the next bilateral meeting with President Putin should take place after the Russia witch hunt is over", referring to U.S. special counsel Robert Mueller's investigation into allegations of Russian interference in the 2016 presidential election and whether there was collusion among the Trump campaign team. "This is outrageous", said Blitzer, a former White House correspondent.
He went on to praise Mr Trump, saying he "strives to fulfil the promises, above all, given to voters".
US intelligence agencies concluded previous year that Moscow waged an influence campaign to help Trump win the White House by hacking into Democratic Party computers and using fake social media accounts to denigrate Democratic candidate Hillary Clinton.
On Friday afternoon, Trump will convene a National Security Council meeting devoted to the issue of election security.
A USA special counsel is now investigating whether the Trump campaign colluded with Russian spies to manipulate the vote.