Russian-born Anna Shapiro, 30, and husband Alex King, 42, fell ill at the restaurant on Sunday - with Shapiro claiming to have been poisoned by Russian agents.
Both tested negative for poisoning and have since been discharged from hospital.
The claim follows a state of heightened tension in Salisbury following the Novichok assassination attempt carried out there by Russian agents earlier this year.
Sergei Skripal and his niece Yulia were seriously harmed by the attack in March, while 44-year-old Dawn Sturgess was killed when she accidentally came into contact with the agent in June.
Police are reportedly investigating whether the claim from Shapiro was a hoax - while The Times have claimed Prezzo is preparing to sue the couple, according sources close to the company.
A spokesman for Prezzo denied that the matter had been discussed - but did not rule out future legal action.
He said: "The reality is that the police's investigation is still going on, so this isn't something that would be discussed or considered until the conclusion of their investigation at the earliest."
Nigel Adams, a partner at law firm Goodman Derrick LLP, told The Caterer that if Prezzo has been the victim of a hoax, then it may want to look at a claim for malicious falsehood against the couple involved.
He said: "A claim for malicious falsehood would assert that the couple made a false allegation of poisoning and did so with a view to deliberately causing loss to Prezzo.
"If the restaurant chain has lost bookings and suffered spoiled food as a result of the hoax, they can claim those losses from the hoaxers.
"Prezzo will need to establish the poisoning claim was a hoax and that is not ordinarily easy to do, but here Prezzo might have an advantage as the police will no doubt be looking into this and their findings could help Prezzo establish its case.
"The whole thing could result in these two diners being handed a very large bill for their meal."