The British government has said Putin is ultimately responsible for the attack, a claim the Kremlin has strongly denied.
"I hope they will turn up themselves and tell about themselves", he said in comments that hinted they will make a public statement shortly.
Alexander Petrov, left, and Ruslan Boshirov have been charged by British police with attempted murder.
Russian Federation has repeatedly denied any involvement in the incidents.
The comments came one week after United Kingdom authorities charged two Russian men with using a Novichok nerve agent to poison former KGB spy Sergei Skripal and his daughter Yulia.
"We know who they are".
Britain has said the two suspects were Russian military intelligence officers nearly certainly acting on orders from high up in the Russian state.
In a statement that deepened the diplomatic crisis between the two countries, the Prime Minister said: 'The GRU is a highly disciplined organisation with a well-established chain of command.
Citing the GRU's reputation for strict discipline and hierarchy, May said of the attack on the Skripals, "It was nearly certainly also approved outside the GRU, at a senior level of the Russian state". He added they would contact the media and tell their story soon.
The Kremlin leader spoke amid new revelations indicating that the "spies" who planted novichok in Salisbury may have stolen the identities of Russian people for their mission. "There is nothing particularly criminal there, I assure you", he said.
Police are linking the attack to a separate Novichok poisoning on 30 June, when Dawn Sturgess and Charlie Rowley became unwell at a house in Amesbury, about eight miles away.
The Russian Foreign Ministry has previously rejected May's accusations, noting both the unacceptable tone of her statement and the necessity for a thorough and unbiased investigation which has been so far made impossible by London's unwillingness to cooperate with Moscow.
An Interpol Red Notice and European arrest warrant have been issued, though the Russian Constitution forbids them from being extradited.
Ms Sturgess died in hospital in July, just over a week after the pair fell ill.