"Trump proposed holding a meeting at the White House in Washington," Putin's top foreign policy aide, Yury Ushakov, told reporters.
On March 20, Trump called Putin to congratulate him on his re-election, and the US leader told reporters afterwards that the two would "probably get together in the not-too-distant future."
In calling Putin, Trump ignored explicit advice from his national security advisers not to do so, The Washington Post has reported, quoting officials familiar with the call.
Ushakov said on Monday, however, that the two sides have not had any "concrete discussions" about the summit since that rare Trump-Putin phone conversation.
"It was Trump himself who proposed holding the meeting," Ushakov said.
He expressed the hope that Russia and the United States could return to "constructive and serious dialogue."
After the call, Washington expelled 60 Russian diplomats and shut down a Russian consulate in Seattle, joining Britain's allies in responding to the poisoning of former double agent Sergei Skripal and his daughter Yulia in the English city of Salisbury on March 4.
Moscow responded by sending home 60 US diplomats and closing Washington's consulate in Saint Petersburg.
Washington has, however, said Russia is free to apply to accredit more diplomats to replace those expelled.