The meeting, in the western Japanese city of Osaka, will take place on Friday, said Putin's top foreign policy advisor Yury Ushakov. May's office declined to confirm the talks.
Earlier this month, Putin said it was time to "turn the page" on the countries' difficult relationship after the poisoning of a former Russian spy in the English city of Salisbury last year.
Ushakov said Moscow would welcome a breakthrough in relations with London.
"The leaders will discuss sensitive issues, and a lot of them have accumulated in our ties," he told reporters.
"If any opportunities can be found to build cooperation anew we would only welcome this."
The Skripal affair
The two leaders last met for formal talks on the sidelines of a G20 summit in China in 2016.
But the 2018 poisoning of the former Russian spy and double agent Sergei Skripal, which the UK has blamed on Russia, led to dozens of tit-for-tat diplomatic expulsions and a breakdown in already fragile ties between Moscow and London.
Russia has repeatedly denied any part in his killing.
Earlier this month, in a clear sign he was ready to move on, Putin said: "Global issues related to national interests in the economic and social spheres and global security are more important than games of security services."
London said at the time that a new relationship would only be possible following changes from Moscow.
Britain is waiting to hear who the country's next leader will be as May prepares to step down from power.
The name of a new British prime minister is expected to be announced on July 23, and former foreign minister Boris Johnson is the favourite in the race.
Putin has expressed hope that relations could improve under a new prime minister in Britain, suggesting the new leader should think about British companies operating in Russia.
Economic ties between the two countries have suffered over the past year. Trade turnover between January and April dropped by almost 15 percent to $4 billion, the Kremlin said.
"On the whole, large British companies who are among world leaders in their industries continue to be consistently in favour of strengthening cooperation with Russian partners," said Ushakov.
Skripal sold secrets to Britain and moved there after a 2010 spy swap.
Last year, he and his daughter were poisoned with a military-grade nerve agent, the first use of chemical weapons in Europe since World War II.
London says the attempted assassination was "almost certainly" approved by the Russian state.
The case has strong echoes of the poisoning of former Russian agent Alexander Litvinenko in Britain in 2006.