Wiltshire Police said they were called by the ambulance service at about 6.45pm on Sunday after a "medical incident" involving a man and a woman at Prezzo restaurant.
South Western Ambulance Service said it was called to the scene more than an hour earlier and dispatched four ambulances including a hazardous area response team.
Roads were cordoned off around the Italian restaurant as police tried to determine what caused the couple to fall ill.
Pictures from local press showed a hazardous material suit in the area while a police officer was seen in protective clothing with their face covered.
Public Health England was informed of the incident, while firefighters were also at the scene.
Witness Sam Proudfoot, 16, said he saw a person in a hazardous material suit go between the restaurant and the ambulance.
"I've been told two people were taken ill in there," the student said.
"There's a man in a full white body suit with a mask to his mouth going in and out of the back of the ambulance and the restaurant.
"The police were interviewing a couple of dozen people."
A witness in the nearby Cafe Rouge restaurant said they could see police bagging something up in the street outside.
A spokeswoman for the ambulance service said the patients were conscious at the last update and were being treated at the scene.
Tensions are heightened in the cathedral city because of recent Novichok poisonings.
The Italian restaurant is a short walk from Queen Elizabeth Gardens, which was until recently closed off after 44-year-old Dawn Sturgess was fatally poisoned by the nerve agent.
Also nearby is Zizzi, the Italian restaurant where former spy Sergei Skripal and his daughter Yulia visited before they were taken ill.
Police said Sturgess was killed by the same chemical used in an alleged hit by Russian military intelligence officers on Skripal.