May made this known before she met Russian President Vladimir Putin on Friday.
Britain has accused two officers from Russia’s military intelligence agency, the GRU, of being responsible for the poisoning of Sergei Skripal and his daughter Yulia with a nerve agent in the English city of Salisbury in March 2018.
May met Putin on the sidelines of a G20 summit in Osaka, Japan.
“Russia does not allow the extradition of its nationals, but European arrest warrants are out for those two individuals.
“If they set foot outside Russia, we will be making every effort that they are brought to justice,’’ she told the BBC before the meeting.
She urged Moscow to stop its ‘destabilising activities’, including disinformation and cyber attacks.
Putin rejected Britain’s claims of Russian state involvement in the nerve agent poisoning, suggesting that both sides could be ready to put the incident behind them.
“I think Russia and UK are both interested in fully restoring our relations, at least I hope a few preliminary steps will be made,’’ he told the Financial Times.
Putin added that spies, who betray their country, must be punished.