But their governments did not have time to arrange details before the US expelled 60 Russian diplomats following the Salisbury poisoning, they revealed.
Moscow's ambassador to the US Yuri Ushakov made the claims on Monday, adding he hoped for a "serious and constructive dialogue" between the two countries.
White House Press Secretary Sarah Huckabee Sanders confirmed Mr Trump invited Mr Putin to Washington during the call.
But she stressed the White House was among "a number of potential venues" discussed as a location for the bilat.
She cited an announcement hours after the phone call on 20 March, that said the two leaders had discussed meeting in person.
Ms Huckabee Sanders told newsmen"as the President himself confirmed on March 20, hours after his last call with President Putin, the two had discussed a bilateral meeting in the 'not-too-distant future' at a number of potential venues, including the White House."
Some politicians from Mr Trump's Republican party had criticised him for making the congratulatory call to Mr Putin.
Senator John McCain said at the time: "An American president does not lead the Free World by congratulating dictators on winning sham elections.
"And by doing so with Vladimir Putin, President Trump insulted every Russian citizen who was denied the right to vote in a free and fair election."
While Senate majority leader Mitch McConnell said that calling Mr Putin "wouldn't have been high on my list".
Mr Trump defended his decision on Twitter, saying President Barack Obama did the same in 2012.
He wrote on Twitter: "the Fake News Media is crazed because they wanted me to excoriate him. They are wrong! Getting along with Russia and others is a good thing, not a bad thing."