The statement was part of an "information and political campaign based on provocation," said Russian foreign ministry spokeswoman Maria Zakharova in comments carried by news agencies.
"This is a circus show in the British parliament," she added.
"Rather than think up new fairy tales, maybe someone in the kingdom could explain how the previous ones ended up — about Litvinenko, Berezovsky, Perepilichny and many others who have mysteriously died on British soil," Zakharova said.
Former spy Alexander Litvinenko, oligarch Boris Berezovsky and whistleblower Alexander Perepilichny all died in the UK in recent years.
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The Kremlin has long rejected any involvement in their deaths, despite a British judge concluding Russian President Vladimir Putin likely approved Litvinenko's killing.
May on Monday pointed the finger at Russia for trying to kill Sergei Skripal, who sold secrets to Britain and later moved there in a 2010 spy swap.
But she stopped short of announcing retaliatory measures against Moscow and instead gave the Kremlin until the end of Tuesday to explain a nerve agents programme allegedly involved in the Skripal case.
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Andrei Lugovoi, a prime suspect in Litvinenko's 2006 killing who has never been extradited and is now a member of the Russian parliament, called May's speech "irresponsible".
"The hasty statements of Theresa May, especially in connection to the nerve agent, allegedly made in Russia, are irresponsible to say the least," he told the Interfax news agency.
"Soon everything will be classified, as this is the only way to make sure the public has no access to the case materials, as was the case in the Litvinenko affair," he said.
FSB General Vladimir Dzhabarov, who serves on the international affairs committee of Russia's upper house, told agencies the poisoning of Skripal could have been carried out by Britain or a third country with the aim of "blaming and blackening Russia".
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