The “strength of the support” shown to the UK over the poisoning of former Russian agent Sergei Skripal and his daughter has been encouraging, the foreign secretary has said.
Boris Johnson said the UK had been talking to friends and there had been a “willingness” to show “solidarity”.
Russia’s foreign minister said claims of Russian involvement were “rubbish”.
The UK has given Russia a midnight deadline to explain why a Russian-made nerve agent was used in the attack.
Russian Foreign Minister Sergei Lavrov said that Russia had been refused access to the substance that was used to poison Mr Skripal and it would not respond to the ultimatum until it was given access.
The UK ambassador to Russia, Laurie Bristow, has been summoned to the Russian Foreign Ministry, according to Russian news agency Interfax.
Former double agent Mr Skripal, 66, and his daughter, Yulia, 33, were found slumped on a bench in Salisbury city centre in Wiltshire on 4 March. 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.
Nato secretary-general Jens Stoltenberg said: “The UK is a highly valued ally, and this incident is of great concern.”
US Secretary of State Rex Tillerson said it appeared the “really egregious act… clearly came from Russia” and there should be “serious consequences”.
Home Secretary Amber Rudd will chair a meeting of the government’s emergencies committee Cobra this morning to discuss the case.
Mrs May told the Commons on Monday that the poison used in the attack was a military-grade nerve agent developed by Russia. She said it was part of a group of nerve agents known as Novichok.
“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”.
Conservative minister Dominic Raab said sanctions could be wide-ranging but played down the involvement of Nato.
He told Radio 4’s Today programme that Mrs May “chose her words very carefully” in her Commons statement on the incident.