Wiltshire police have confirmed that the couple were not exposed to any nerve agents and were not linking the incident to the attacks on Salisbury and Amesbury.
"Anyone with information should call 101 and quote long number 271 of 16 September", the force posted on Twitter.
Anna became an Israeli citizen in 2006 - against her family's wishes.
Police investigating a suspected poisoning at a restaurant in Salisbury on Sunday are expected to consider whether it was a hoax.
Wiltshire police have not commented on whether the incident was a hoax but told ABC News they could not "rule anything in or out" until they spoke to King.More news: Ticketmaster Recruiting Its Own Scalpers To Squeeze More Money Out Of Fans
The local police department said the pair who got sick on Sunday were tested for traces of the deadly nerve agent in light of "recent events in the south of the country".
Police sources told United Kingdom reporters that they want to question Alex King, 42, and his wife Anna Shapiro, 30, who were hospitalized on Sunday after reportedly falling ill at a restaurant in Salisbury.
The Sun's original story about Shapiro's claims was unavailable on its website on Thursday morning.
Prezzo in Salisbury was on lockdown after a man and a woman fell ill.
The Eton-educated businessman is said to have sneaked into a line-up of celebrities at a premiere for the film "The History Boys" in London.More news: North Korea summit: Kim pledges nuclear cuts if Americans follow suit
He claimed the "prank" was carried out as part of a £100,000 bet with his employer - the convicted fraudster Edward Davenport.
Salisbury District Hospital said "both patients are now medically fit" and have been discharged from hospital.
In 2004, he pleaded guilty to three counts of distributing indecent photographs or pseudo-photographs of children at London's Horseferry Road Magistrates' Court.
Officers will question Alex King, 42, who was admitted to hospital after apparently becoming unwell at Prezzo, an Italian restaurant near where Sergei Skripal and his daughter Yulia were poisoned with novichok in March.
"While this incident did not involve nerve agents, it was still clearly very stressful for our staff who, once again, demonstrated the very best of the NHS".More news: Audi e-tron SUV officially sees the light of day