Sergei Skripal, 66, was found unconscious in Salisbury, Wiltshire, along with a woman in her 30s, reported by the BBC to be his daughter, Yulia, shortly after 4pm on Sunday.
As CCTV believed to show the pair in the moments before they were found slumped on a bench emerged, the UK’s top counter-terrorism officer, Metropolitan Police Assistant Commissioner Mark Rowley, said: “We have to be alive to the fact of state threats.”
Russia said it had no information regarding the “tragic situation” but is willing to co-operate, as comparisons were drawn with the death of dissident Alexander Litvinenko in 2006.
Andrei Lugovoi, the man suspected of poisoning Mr Litvinenko, dismissed the reports as propaganda.
Mr Skripal was convicted in 2006 of passing state secrets to MI6 before being given refuge in the UK as part of a spy swap.
The former colonel in the Russian military intelligence, who was sentenced to 13 years in prison, was among four convicts who were given pardons and one of two sent to Britain in 2010 in a deal that was said at the time to be the largest exchange since the Cold War.
The Castle Street branch of pizza chain Zizzi was closed late on Monday night “as a precaution” as part of the police investigation.
A CCTV image of a man and woman walking steadily through an alleyway between the Zizzi restaurant and the bench near a shopping centre where Mr Skripal was found is believed to be of interest to police.
An image of the pair - shot at 3.47pm on Sunday - was taken by officers from a camera at Snap Fitness 24/7, the gym manager told the Press Association.
Cain Prince, 28, said: “Police had a good look at the footage and were interested in these two people. It was the only image they took away.
“They wanted a list of everyone in the gym between 3pm and 4pm as well.”
Freya Church, 27, who spotted the pair “slumped” and “passed out”, said the couple in the CCTV image were “100%” the people she saw on Sunday.
The gym worker, from Salisbury, said: “She was leaning on him, slumped. She looked passed out and he was looking up doing these hand movements.
“His eyes were glazed. To be honest I thought they were just homeless.”
Wiltshire Police said officers were as yet “unable to ascertain” whether the pair, who are both in a critical condition in intensive care, have been victims of a crime.
Shadow home secretary Diane Abbott cautioned against speculation but said the incident “does bear a striking similarity” to the death of Mr Litvinenko, who was poisoned after drinking tea laced with radioactive polonium-210.
She told BBC Radio 4’s Today programme she would write to Home Secretary Amber Rudd to ask for assurances “if it does prove to be the case that the Russian state is involved”.
“I don’t like defaulting to a ‘red menace’ analysis, but we can’t allow London and the Home Counties to become a kind of killing field for the Russian state and its enemies,” she said.
Anti-terror chief Mr Rowley said his specialists were supporting the investigation.
He said: “If you look back at other cases like (Alexander) Litvinenko, if necessary we will bring that investigation into the counter-terrorism network.
“At the moment the key is, though, to get to the bottom of what caused this.”
Asked about a series of suspicious Russian-linked deaths in the UK, Mr Rowley added: “There are deaths which attract attention.
“I think we have to remember that Russian exiles are not immortal, they do all die and there can be a tendency for some conspiracy theories.
“But likewise we have to be alive to the fact of state threats as illustrated by the Litvinenko case.”
Kremlin spokesman Dmitry Peskov said Russia did not “have any information” and had not been approached for help in the investigation.
“Moscow is always open to co-operation,” he added.
Temporary Assistant Chief Constable Craig Holden, of Wiltshire Police, said: “The pair, who we believe are known to each other, did not have any visible injuries and were taken to Salisbury District Hospital.
“This has not been declared as a counter-terrorism incident and we would urge people not to speculate. However, I must emphasise that we retain an open mind and we will continue to review this position.”
Igor Sutyagin, who was part of the same swap deal as Mr Skripal and is now a research fellow at RUSI, also urged caution.
He told the Associated Press: “There are lots of former security officers that deserted to the West. It is necessary to balance this information.”
Salisbury District Hospital declared a major incident but told patients to attend appointments as normal unless advised otherwise.
A spokesman for Public Health England (PHE) said anyone exposed to the unknown substance had been decontaminated “as is standard practice in situations like this”.