People who visited the Olivello restaurant between August 25 and 29 are being urged to self-isolate as a precaution.
It follows three positive cases being linked to the Linthorpe Road restaurant.
The council has issued guidance saying that the self-isolation period for anyone affected should begin from the date they were at the restaurant.
Only those who were displaying symptoms should go for a test, the council added.
Anyone with symptoms can book a free test online or by calling 119.
A mobile testing site will be at Newport Community Hub until Thursday. The permanent Middlehaven site is open from 8am-8pm daily.
The council said that Olivello management is co-operating fully and is taking responsible actions.
A deep clean has taken place although the restaurant will be closed until Monday when the situation will be reviewed.
Attempts will be made to contact customers directly to provide the urgent self-isolation advice.