Russia is investigating the attempted murder of a contact of the former KGB agent, Alexander Litvinenko, who died in London last month.
Mr Kovtun has been interviewed by British police
Dmitry Kovtun met Mr Litvinenko hours before he fell ill from the radioactive poison that eventually killed him.
Reports that Mr Kovtun is in a critical condition have sparked speculation he too may have been poisoned. A lawyer linked to him denies he is unwell.
The Russian investigation is being run parallel to a British murder inquiry.
The UK's minister for communities and local government, Ruth Kelly, said on the BBC's Question Time programme on Thursday that she expects Russia "to co-operate fully" with the British investigation.
A funeral service for Mr Litvinenko was held at a central London mosque on Thursday.
The former spy, a vocal critic of the Kremlin, had issued a statement on his deathbed accusing Russian President Vladimir Putin of ordering his poisoning.
But the Kremlin has dismissed the allegations.
On Thursday, seven workers at a bar where Mr Litvinenko met Mr Kovtun before falling ill reportedly tested positive for polonium-210, the radioactive substance blamed for his death.
The two men met at the Pine Bar in London's Millennium Hotel on 1 November.
The UK's Health Protection Agency said the risk to the general public was likely to be "very low".
Mr Kovtun was taken ill after he was questioned by British and Russian investigators, Russia's Interfax news agency said on Thursday.
Mr Litvinenko died on 23 November in a London hospital
"By the doctors' diagnosis, Kovtun's condition is critical," the agency said. Interfax had earlier reported Mr Kovtun was in a coma.
However, Andrew Romashov, a lawyer, said Mr Kovtun's condition had not deteriorated since he met the prosecutors.
"Kovtun's condition is satisfactory," he said, dismissing the earlier reports as "provocation".
Mr Romashov, a lawyer for Mr Kovtun's business partner, ex-spy Andrei Lugovoi, said he had been in contact with Mr Kovtun's representatives.
Both Mr Kovtun and Mr Lugovoi are said to have met Mr Litvinenko in the Millennium hotel before he fell ill.
Mr Lugovoi is also in hospital but his condition is unclear.
Russian prosecutors on Thursday announced they would launch a criminal investigation into Mr Litvinenko's death and what they described as the attempted murder of Mr Kovtun.
According to Russia's Itar-Tass news agency, Moscow's investigation will take place "within the framework" of assistance being offered to visiting British investigators.
Prosecutors in Moscow said their criminal investigation would enable them to conduct a trial in Russia - if a Russian suspect was found.
Moscow has already said it will not extradite any suspects to the UK.