This was reported by the Italian daily newspaper "Il Messaggero" on Friday.
However, it is still unclear which private retail chain or wholesaler will be awarded the contract to operate the outlet, which was founded in 1932.
The background to the privatisation is apparently financial bottlenecks in the Vatican.
The supermarket currently employs around 30 people.
As there is no VAT or taxes on alcohol, fuel or tobacco in the Vatican, the price structure is different to that in Italy.
So far, strict care has been taken to ensure that only Vatican employees shop in the "Annona".
An expansion of the clientele could lead to market distortions with regard to the surrounding Italian supermarkets, where a sales tax of 22 per cent has to be paid.