Sergei Skripal, 66, and his daughter Yulia, 33, have been in hospital in a critical condition since March 4 when they were found unconscious on a bench outside a shopping centre in the southern English cathedral city of Salisbury.
May has warned that if Russia was behind the poisoning of Skripal, a former colonel in GRU military intelligence, then Britain will respond robustly. Russia has repeatedly denied any involvement in the attack.
The chairman of the British parliament's foreign affairs committee, Tom Tugendhat, said the attack looked like state-sponsored attempted murder and he expected May to blame Moscow.
"Frankly I would be surprised if she did not point the finger at the Kremlin," Tugendhat told BBC radio.
He said Russia's so-called oligarchs, who amassed fortunes as the Soviet Union crumbled and during President Vladimir Putin's 18-year rule of Russia, should be denied entry to the luxuries of London and the West.
British counter-terrorism police say a nerve agent - usually a small molecule based on phosphorus that interferes with the transmission of nerve signals - was used on Skripal and his daughter. But they have refused to specify which poison was used.
If Russia is publicly accused of carrying out the attack, it would send already strained relations between London and Moscow to a new post-Cold War low, with potential repercussions for some Russian money in Britain.
The Russian Foreign Ministry says London is whipping up anti-Russian hysteria.
The British capital has been dubbed "Londongrad" due to the large quantities of Russian money that have poured in since the collapse of the Soviet Union in 1991.
May last year said Putin was seeking to undermine the West and the international order by meddling in elections. She promised to ensure corrupt money does not flow into Britain from Russia.
Putin, who took over as Kremlin chief from Boris Yeltsin on the last day of 1999, has denied allegations that Russia meddled in the 2016 U.S. presidential election and says the West has repeatedly tried to undermine Russian interests.
Skripal betrayed dozens of Russian agents to British intelligence before his arrest in Moscow in 2004. He was sentenced to 13 years in prison in 2006, and in 2010 was given refuge in Britain after being exchanged for Russian spies.
British Foreign Secretary Boris Johnson also drew parallels between the poisoning of Skripal and the murder of former KGB agent Alexander Litvinenko, who died in London in 2006 after drinking green tea laced with radioactive polonium-210.
A British public inquiry found the killing of Litvinenko had probably been approved by Putin and carried out by two Russians, Dmitry Kovtun and Andrei Lugovoy - a former KGB bodyguard who later became a member of the Russian parliament.
Both denied responsibility, as did Moscow.
After Skripal was found slumped on a bench in Salisbury, health officials in Britain said there was no wider risk to public health.
But some police investigators wore full chemical and biological suits and the army was later deployed to help remove items from the scene.
On Sunday, hundreds of people who visited the city's Zizzi restaurant or the Bishop's Mill pub were told to wash their clothes after traces of the substance used to attack Skripal were found at both sites.
Jenny Harries, deputy medical director at Public Health England, suggested there may be a very small health risk associated with repeated contact with belongings that may have been contaminated.
"Wipe personal items such as phones, handbags and other electronic items with cleansing or baby wipes and dispose of the wipes in the bin," Harries suggested. "Wash other items such as jewellery and spectacles which cannot go in the washing machine with warm water and detergent."
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