Russia has been set a deadline of midnight to reveal what it knows about the attack last week.
Theresa May said on Monday it was "highly likely" that Moscow was responsible for the poisoning in England of Russian former double agent Sergei Skripal and his daughter using a military-grade nerve agent.
May told parliament that either the Russian state was directly responsible for the poisoning or it had allowed the nerve agent to get into the hands of others. London has given Russia until the end of Tuesday to explain its use.
British officials had identified the substance as being part of the Novichok group of nerve agents that were developed by the Soviet military during the 1970s and 1980s, May said.
Skripal, 66, and his 33-year-old daughter, Yulia, have been in hospital in critical condition since being found unconscious on a bench outside a shopping centre in the city of Salisbury on March 4.
"Should there be no credible response, we will conclude that this action amounts to an unlawful use of force by the Russian state against the United Kingdom," she said, calling the attack a "reckless and despicable act".
Russia's foreign ministry hit back immediately, saying May's comments were a "circus show" and part of a political information campaign against Russia.
U.S. Secretary of State Rex Tillerson said the United States had "full confidence" in Britain's assessment that Russia was likely responsible.
"We agree that those responsible - both those who committed the crime and those who ordered it - must face appropriately serious consequences," Tillerson said in a statement.
Earlier, White House spokeswoman Sarah Sanders said the United States stood by America's "closest ally," but she stopped short of blaming Russia for the attack.
NATO Secretary General Jens Stoltenberg said the use of any nerve agent was "horrendous and completely unacceptable" and "this incident is of great concern to NATO".
Relations between Britain and Russia have been strained since the murder in London of former KGB agent Alexander Litvinenko, who died in 2006 after drinking green tea laced with radioactive polonium-210...