Russia has hotly contested the allegations and on Wednesday Putin escalated the war of words by saying Russian officials “know who these people are” but denied the men were members of military intelligence.
He called on the two men to contact the media and “tell about themselves”.
His intervention risks widening the gulf between Russia and the UK over the attempted assassination, which triggered a wave of diplomatic expulsions by both sides.
Three months later Charlie Rowley and Dawn Sturgess, from nearby Amesbury, were hospitalised after coming into contact with Novichok found in a perfume bottle picked up by Rowley.
Sturgess later died and a murder inquiry was launched.
Theresa May said around 250 detectives had trawled through 11,000 hours of CCTV footage to identify the attackers and had taken more than 1,400 statements.
“Working around the clock, they have carried out painstaking and methodical work to ascertain exactly which individuals were responsible and the methods they used to carry out the attack,” she told MPs.
May said the UK would now “step up efforts” against the GRU “specifically”.
She added: “The actions of the GRU are threat to all our allies and all our citizens.”
May’s official spokesman said the two suspects used valid state-issued Russian passports to travel to the UK.
The City Stay Hotel, where the suspects are believed to have stayed in east London, said: “The City Stay Hotel in Bow Road is open for business as usual. We are fully supporting the police investigation. We are reassured that the police and Public Health England have confirmed very clearly there is no health risk whatsoever to our guests or our staff.”
Metropolitan Assistant Commissioner Neil Basu said the two men are aged around 40 and it is likely they were travelling under aliases and Petrov and Boshirov are not their real names.