Sergei Skripal, 66, who moved to Britain in a 2010 spy swap, is in a critical condition in hospital along with his daughter Yulia after they collapsed on a bench outside a shopping centre on Sunday.
A policeman also fell ill after coming to their aid but is showing signs of recovery, according to interior minister Amber Rudd.
"The two targets are still in very serious condition, the policeman is talking and is engaging so I'm more optimistic for him, but it's too early to say," she told ITV1's Good Morning Britain.
On Wednesday, British police confirmed for the first time that a nerve agent was used and that their probe was now an attempted murder investigation.
"Police are now in a position to confirm that their symptoms are a result of exposure to a nerve agent," said the Metropolitan Police.
"Scientific tests by government experts have identified the specific nerve agent used which will help identify the source."
The Times newspaper reported on Thursday that Skripal's condition was particularly severe, quoting a senior unnamed British government official.
Police have cordoned off an Italian restaurant and a pub that the pair are believed to have visited.
A diner in the restaurant told the Times that Skripal had eaten there on Sunday, and that he was in an agitated state.
"He was going absolutely crazy, I didn't understand it and couldn't understand them," the witness said.
"He didn't seem ill physically but perhaps mentally ill with the way he was shouting."
He noted the "echoes" with the 2006 poisoning in London of former Russian spy and Kremlin critic Alexander Litvinenko, which Britain has blamed on Russia.
Defence Secretary Gavin Williamson told British television on Thursday that Russia was "becoming an ever-greater threat". Fellow Conservative MP Nick Boles tweeted that "I do not see how we can maintain diplomatic relations with a country that tries to murder people on British soil."
However, the interior minister called for "cool heads" while the police investigated.
Hundreds of counter-terrorism detectives are working "around the clock" to create a timeline of the victims' movements, with "many hours" of CCTV under review, police said.
The Times reported police are probing whether Skripal's daughter, who arrived in Britain from Moscow last week with "gifts from friends", may have inadvertently brought the nerve agent into the country.
The paper previously said investigators would also examine the 2012 death of Skripal's wife from cancer, and that of his 44-year-old son last year in St Petersburg, reportedly from liver problems.
Multiple British media outlets reported Wednesday that Prince William would now not attend the tournament, citing royal sources.
Skripal was a former colonel in Russian military intelligence who was jailed in his country for betraying agents to Britain's MI6 secret service.
He was pardoned before being flown to Britain as part of a high-profile spy swap involving Russia and the United States in 2010.
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