However Russia issued a warning that any punitive action would elicit a "response".
"Russia will not respond to London's ultimatum until it receives samples of the chemical substance [used in the attack]," a Russian embassy spokesman told Reuters. Russia was not involved in the nerve attack, the spokesman said.
In the meantime, counter-terror police in London have launched an investigation into the unexplained death of Russian businessman Nikolai Glushkov in London on the weekend. Glushkov, a critic of Russian President Vladimir Putin, had sought political asylum in 2006. Police said there was no evidence to suggest a link to the Salisbury attack.
In a speech to the House of Commons on Monday, May said “extensive measures” would follow if the Kremlin failed to rebut her conclusion that Russia was responsible for the attempted murder that had put innocent lives at risk.
On Tuesday, May spoke to US President Donald Trump, who said, “it sounds to me like it would be Russia (behind the attack) based on all the evidence”.
The White House issued a statement saying the US "stands in solidarity with its closest ally and is ready to provide any assistance the United Kingdom requests for its investigation.
"The two leaders agreed on the need for consequences for those who use these heinous weapons in flagrant violation of international norms."
A Downing Street spokesperson said after the call that the US was "with the UK all the way”.
May also spoke to German Chancellor Angela Merkel who, the spokesperson said, “condemned the attack and said she stood in full solidarity with the UK”.
“They discussed the pattern of aggressive Russian behaviour and agreed it would be important to act in unison with allies to counter it.”
French President Emmanuel Macron reportedly also expressed solidarity with the UK.
Senior European parliamentarian Guy Verhofstadt called for EU leaders to agree to “a common European response to this outrage” at a summit next week.
Also on Tuesday:
A bill before Parliament would enable officials to refuse entry to the UK to anyone known to have been involved in human rights abuses. The government has opposed it until now: it may change its mind, if it could be put to anti-Russian use.
In January, "Unexplained Wealth Orders" came into force – a new investigative power that allows the seizure of suspected corrupt assets.