It will be the first time Wolverhampton has hosted a match involving the England senior men’s team since December 1956.
England were ordered by Uefa to play one senior men’s match without fans as a punishment for the unrest last June.
As well as Italy and Hungary, Southgate’s side have also been drawn alongside Germany in Nations League Group A3.
The Three Lions start with a trip to Hungarian capital Budapest on Saturday, 4 June (17:00 BST) before taking on Germany in Munich on Tuesday, 7 June (19:45 BST).
The Nations League group stages will be completed in September before the 2022 Fifa World Cup finals in Qatar begin in November.