US President Donald Trump has said there was a high probability of a summit with his Russian counterpart Vladimir Putin in the near future.
"There's a good chance that there will be a meeting very soon," Trump told reporters, after the White House confirmed he was open to a meeting with Mr Putin to discuss Russia's invasion of Ukraine.
Earlier, the New York Times reported that Mr Trump and Mr Putin could meet as early as next week, citing two people familiar with the plan.
Mr Trump then plans to meet with Mr Putin and Ukrainian President Volodymyr Zelensky, the newspaper reported, adding that the plans were disclosed in a call with European leaders.
Mr Trump, who promised to end Russia's war in Ukraine on "day one" during his presidential campaign, has held several phone calls with Mr Putin and has met with Mr Zelensky since returning to the White House in January.
However, in recent weeks, he has become increasingly frustrated with Moscow over a lack of progress towards ending the three-year conflict.
Kremlin foreign policy aide Yuri Ushakov said talks between Russian President Vladimir Putin and US special envoy Steve Witkoff were "useful and constructive".
Mr Witkoff held around three hours of talks with Mr Putin in the Kremlin, two days before the expiry of a deadline set by President Donald Trump for Russia to agree to peace in Ukraine or face new sanctions.
Mr Ushakov told Russian news outlet Zvezda that the two sides discussed the conflict in Ukraine and the potential for improving US-Russia relations.
He said Moscow had received certain "signals" from Mr Trump and had sent messages in return.
Mr Trump has warned that he would impose sanctions on Russia if Moscow does not agree to a ceasefire in the war in Ukraine before Friday
The measures would include heavy tariffs on countries that buy its oil, of which the biggest are India and China.
However, Mr Putin is unlikely to bow to the ultimatum because he believes he is winning the war and his military goals take precedence over his desire to improve relations with the US, three sources close to the Kremlin said.
"The visit of Witkoff is a last-ditch effort to find a face-saving solution for both sides. I don't think, however, that there will be anything of a compromise between the two," said Gerhard Mangott, an Austrian analyst and member of a group of Western academics and journalists who have met regularly with Mr Putin over the years.
"Russia will insist it is prepared to have a ceasefire, but (only) under the conditions that it has formulated for the last two or three years already," he said.
"Trump will be under pressure to do what he has announced - to raise tariffs for all the countries buying oil and gas, and uranium probably as well, from Russia."
The Russian sources said that Mr Putin is sceptical that yet more US sanctions will have much of an impact after successive waves of economic penalties during three and a half years of war.
Watch: Trump 'not happy' with Putin but remains committed to diplomatic solutions
The Russian leader does not want to anger Mr Trump, and he realises that he may be spurning a chance to improve relations with Washington and the West, but his war goals are more important to him, two of the sources said.
Mr Witkoff, a real estate billionaire, has had several long meetings with Mr Putin. He had no diplomatic experience before joining Mr Trump's team in January, and critics have portrayed him as out of his depth when pitched into a head-to-head negotiation with Russia's paramount leader for the past 25 years.
On his last visit in April, Mr Witkoff - unaccompanied by diplomats or aides - cut a lonely figure when seated across the table from Mr Putin, Kremlin foreign policy aide Yuri Ushakov and Russian investment envoy Kirill Dmitriev.
Critics have at times accused Mr Witkoff of echoing the Kremlin's narrative.
In an interview with journalist Tucker Carlson in March, for example, Mr Witkoff said there was no reason why Russia would want to absorb Ukraine or bite off more of its territory, and it was "preposterous" to think that Mr Putin would want to send his army marching across Europe.
Ukraine and many of its European allies say the opposite.
Mr Putin denies any designs on NATO territory, and Moscow has repeatedly cast such charges as evidence of European hostility and "Russophobia".
Russia has frequently called on Ukraine to effectively cede control of four regions that Russia claims to have annexed, a demand Ukraine has called unacceptable.
Mr Putin also wants Ukraine to drop its ambitions to join NATO.
The visit comes after Mr Trump said that two nuclear submarines, he deployed following an online row with former Russian president Dmitry Medvedev, were now "in the region".
Mr Trump has not said whether he meant nuclear-powered or nuclear-armed submarines.
He also did not elaborate on the exact deployment locations, which are kept secret by the US military.
Russia, in its first comments on the deployment, urged "caution".
"Russia is very attentive to the topic of nuclear non-proliferation," Mr Peskov said.
"And we believe that everyone should be very, very cautious with nuclear rhetoric," he added.
