Officers, who described it as a “medical incident”, were called by the ambulance service at about 6.45pm.
Roads in the city have been cordoned off, Wiltshire Police said.
The patients were conscious at the last update and being treated at the scene, a spokeswoman for South Western Ambulance Service said.
The restaurant, on the High Street in Salisbury, is a short walk from Queen Elizabeth Gardens, which was closed off after 44-year-old Dawn Sturgess was fatally poisoned by the nerve agent novichok at the end of June.
That was the same chemical used to target former Russian spy Sergei Skripal and his daughter Yulia.
Medics were called at 5.38pm, and four ambulances were sent including a hazardous area response team.
Witness Sam Proudfoot, 16, said he saw 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”.
Wiltshire Police said: “Police were called by the ambulance service to Prezzo, High Street, Salisbury at 6.45pm today following a medical incident involving two people.
“As a precautionary measure, the restaurant and surrounding roads have been cordoned off while officers attend the scene and establish the circumstances surrounding what has led them to fall ill.”