The pair, in their 40s, are believed to have attended a church event hours before they were found unconscious in a property in Muggleton Road, Amesbury, on Saturday.
It's 13 kilometres from Salisbury, where Sergei and Yulia Skripal were poisoned on March 4.
Police did not name the couple, but friends have identified the man as Charlie Rowley and the woman as Dawn Sturgess.
Wiltshire police said they initially suspected the two people had fallen ill after using "possibly heroin or crack cocaine from a contaminated batch of drugs".
Britain accused Russian Federation of poisoning Skripal with Novichok nerve agent, the first known offensive use of such a chemical weapon on European soil since World War II.
Scotland Yard's counter-terror chief Neil Basu said, however, it was too early to confirm whether the incident was linked to the Salisbury attack.
Andrea Sella, professor of inorganic chemistry at University College London, said Novichok nerve agents "are created to be quite persistent - they hang around in the environment, neither evaporating or decomposing quickly".
A major incident has been declared in Wiltshire, southern England, after the mystery substance left two people fighting for their lives, the UK Telegraph reports.
She said: "I understand that those in Salisbury and in surrounding areas will be concerned at this news, particularly those who recently visited areas now cordoned off by police".
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The most important factor for her survival is probably how long her own CDU lawmakers stand by her, and for now they are. He told Sky News: "The problem is, the agreement only exists between the CSU, Seehofer's party and Angela Merkel's CDU".
Paramedics were called on Saturday morning after the woman collapsed and returned later in the day when the man also fell ill.
After being found unconscious in the street, the two spent weeks in critical condition at the hospital.
"We are very aware of the public interest in this investigation and the concern it is understandably causing within our communities", the Wiltshire Police statement said.
The unexplained poisoning of two British citizens with no immediately apparent link to Russian Federation has raised public health concerns in the Salisbury area, where a massive decontamination effort took place after the Skripals were found to have been poisoned with Novichok.
Moscow has rejected British accusations of involvement in the Skripal poisoning, which sparked a diplomatic crisis that saw Russian Federation and the West expelling dozens of diplomats in tit-for-tat moves.
He said he was unaware of any official British requests to Russian Federation for help in relation to the investigation into the new incident and said the Kremlin was concerned by reports that a nerve agent had once again been used on European soil. "This must be done with great care, as you will appreciate, to ensure there is no outstanding risk to both those fearless officers and the public", he said.
Wiltshire Police and Crime Commissioner Angus Macpherson said there was "no reason to think it's connected with matters of last month".
A composite photo of Sergei and Yulia Skripal.