Police declared a major incident on Tuesday after the pair were found unconscious in a house in Muggleton Road on Saturday night.
They are both being treated at Salisbury District Hospital.
The town of Amesbury is roughly eight miles from the city of Salisbury, where former Russian spy Sergei Skripal and his daughter Yulia were targeted with the nerve agent novichok in March.
Police initially believed the Amesbury pair, who have not been named, fell ill after using heroin or crack cocaine from a contaminated batch of drugs.
But further tests are now taking place to establish the substance, with police said to be keeping an open mind as to the circumstances around the case.
It remains unclear if a crime has been committed.
Parts of Amesbury and Salisbury thought to have been visited by the pair will be cordoned off overnight as a precautionary measure.
Those living in the areas have been told to expect an increased police presence.
Salisbury District Hospital is open as usual and patients are advised to attend appointments unless told otherwise.
A Public Health England (PHE) spokesman said The current advice from PHE England, based upon the number of casualties affected, is that it is not believed that there is a significant health risk to the wider public.
This will be continually assessed as further information becomes known.
Mr Skripal and his daughter both survived being poisoned with a nerve agent in an attack the British government has blamed on Russia.
Ms Skripal said after being released from hospital We are so lucky to have both survived this attempted assassination.
Our recovery has been slow and extremely painful.
Her father, who has not yet spoken to the media, was discharged from hospital in May.
The pair each spent weeks in a coma after they were found frothing at the mouth on a bench in Salisbury on March 4.
News Source SkyNews