The meeting, on the margins of a G20 summit in the eastern Chinese city of Hangzhou, was "good, constructive and open," Kremlin spokesman Dmitry Peskov
"Mr Putin gave a clear and unequivocal indication that we are interested in completely re-establishing our relations and dialogue in all areas, including the most sensitive ones," he added.
In that regard Russia is ready "to move forward as far as the British are prepared to go," Peskov said.
British relations with Russia have soured in recent years, notably over Britain's efforts to prosecute the case of Kremlin critic and former spy Alexander Litvinenko, who was murdered by radiation poisoning in London in 2006.
Britain has also been one of the most fervent supporters of Western sanctions against Moscow over Russia's role in the Ukraine crisis.
When May became prime minister in July, following the resignation of David Cameron , Putin said he was ready for "constructive dialogue" with the new British leader.
The Kremlin strongman had accused the British government of being "overconfident" and "superficial" in the June referendum that saw the UK vote to split from the European Union, a decision which cost Cameron his job.
Putin warned that the move to leave the EU "will have consequences for the United Kingdom, for all of Europe and for us, of course."
Many observers have said that Brexit could play into Putin's hands as he has been accused of trying to drive a wedge between EU members.