As reported by Censor.NET citing BBC, Mrs May said if there is no "credible response" by the end of Tuesday, the UK would conclude there has been an "unlawful use of force" by Moscow.
The chemical used in the attack, the PM said, has been identified as one of a group of nerve agents known as Novichok.
Mrs May said: "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 Foreign Secretary Boris Johnson had told the ambassador Moscow must provide "full and complete disclosure" of the Novichok programme to international body the Organisation for the Prohibition of Chemical Weapons.
Mrs May said the UK must stand ready to take much more extensive measures, and these would be set out in the Commons on Wednesday should there be no adequate explanation from Russia.
Retired military intelligence officer Mr Skripal, 66, and his daughter, Yulia, 33, were found slumped on a bench in Salisbury city centre on Sunday 4 March. They remain in a critical but stable condition in hospital.
Det Sgt Nick Bailey, who fell ill attending the pair, remains seriously ill, but has been talking to his family.
Mr Skripal was convicted by the Russian government of passing secrets to MI6 in 2004, but given refuge in the UK in 2010 as part of a "spy swap".
Home Secretary Amber Rudd will chair a meeting of the government emergency committee Cobra on Tuesday to discuss the latest developments in the case.
Addressing the Commons following a meeting of the government's National Security Council, Mrs May said: "This attempted murder using a weapons-grade nerve agent in a British town was not just a crime against the Skripals.
"It was an indiscriminate and reckless act against the United Kingdom, putting the lives of innocent civilians at risk."
She told MPs the positive identification of this chemical agent was made by experts at the UK's Porton Down laboratory.
She said Russia has previously produced the agent and would still be capable of doing so.
The decision to point the finger at Moscow was also based on "Russia's record of conducting state-sponsored assassinations and our assessment that Russia views some defectors as legitimate targets for assassinations", the PM added.
Labour leader Jeremy Corbyn said "robust dialogue" with Russia was needed to avoid escalating tensions further - but he was heckled by Tory MPs when he raised questions about Russian oligarchs donating money to the Conservatives.
Secretary of State Rex Tillerson said the US agreed with the UK that Russia was likely to be behind the attack.
"We agree that those responsible - both those who committed the crime and those who ordered it - must face appropriately serious consequences," he added.
"We stand in solidarity with our allies in the United Kingdom and will continue to coordinate closely our responses."
Mrs May spoke to French President Emmanuel Macron on Monday and "discussed the wide pattern of aggressive Russian behaviour and agreed that it would be important to continue to act in concert with allies to address it", her spokesman said.
Nato Secretary General Jens Stoltenberg said the use of any nerve agent was "horrendous and completely unacceptable" and officials were in touch with the UK.
Downing Street said the incident was not an "article five" matter - a reference to Nato rules which say an attack on one member constitutes an attack on all.
However, the former UK National Security Adviser Lord Ricketts said action would be more effective with a "broader, Nato-EU solidarity behind us".
He added: "We can't out-punch Putin... But we can take a stand and we can invite others to join us."
Russian Foreign Ministry spokeswoman Maria Zakharova said Mrs May's statement was "a circus show in the British parliament".
"The conclusion is obvious - it's another information and political campaign based on provocation," she said.
Earlier, asked whether Russia was to blame, President Vladimir Putin told the BBC: "Get to the bottom of things there, then we'll discuss this."
BBC political editor Laura Kuenssberg says the tone of the meeting between Boris Johnson and the Russian ambassador Alexander Yakovenko was "cool but firm".
She says the men did not shake hands and the foreign secretary expressed the "outrage" of the British public.
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Source: https://en.censor.net.ua/n3055137Source: https://en.censor.net.ua/n3055137
