Sergei Skripal, 66, and his daughter Yulia, 33, are in a critical condition in hospital.
Moscow faces a deadline of midnight tonight to explain why a Russian-made nerve agent was used in the poisoning of former Russian agent Sergei Skripal and his daughter.
The PM said it was "highly likely" Russia was responsible for the attack in Salisbury, Wiltshire, last Sunday.
US Secretary of State Rex Tillerson said it appeared the "really egregious act. . . clearly came from Russia" and there should be "serious consequences".
Moscow called the claims "unfounded".
Home Secretary Amber Rudd will chair a meeting of the government's emergencies committee Cobra later to discuss the case.
Former double agent Mr Skripal, 66, and his daughter, Yulia, 33, were found slumped on a bench in Salisbury city centre. They remain in a critical but stable condition in hospital.
Det Sgt Nick Bailey, who fell ill attending to the pair, remains seriously ill, but has been talking to his family.
Mrs May told the Commons on Monday that the poison used in the attack was Novichok - a military-grade nerve agent developed by Russia.
"Either this was a direct action by the Russian state against our country, or the Russian government lost control of its potentially catastrophically damaging nerve agent and allowed it to get into the hands of others," she said.
Mrs May said the Foreign Office had summoned Russia's ambassador to "explain which of these two possibilities it is".
She warned that if there was no "credible response" by the end of Tuesday, the UK would conclude there has been an "unlawful use of force" by Moscow.
She said the UK must "stand ready to take much more extensive measures" against Russia than it had previously.
She said these measures would be set out in the Commons on Wednesday should there be no adequate explanation from Russia.
She described the poisoning as "an indiscriminate and reckless act against the UK, putting the lives of innocent civilians at risk".