The match will take place at the Britannia Stadium on Saturday August 10, just one week before the Potters begin their Premier League campaign with a trip to Liverpool.
"We were approached by a number of European clubs about the possibility of playing at the Britannia Stadium once we made it clear we would be staging a pre-season game at home," chief executive Tony Scholes told the club's official website.
"Mark Hughes felt that Genoa would provide the right test at this stage of our build-up."
Genoa finished 17th in Italy's top division last season and avoided relegation by just six points.
Stoke have not played a first-team friendly at the Britannia Stadium since 2009.