Zirkzee came to Bayern Munich in the summer of 2017 but has not established himself in the first team of the Bavarian giants.
He spent the previous season on loan with Belgian side Anderlecht and after impressing for them has garnered Premier League interest.
Both Fulham and Southampton are interested in the Bayern Munich striker, but now new competition has arisen for the Premier League duo.
Stuttgart are interested in adding the young striker to their squad this summer should they face a key loss, according to Sky Deutschland.
The German club are fearing losing Sasa Kalajdzic this summer, who has been linked with Manchester United and Everton, and will pursue Zirkzee if that happens.
Stuttgart have had a talk with Bayern Munich on the phone regarding Zirkzee, although at this stage there is no agreement between the player and them.
Bayern Munich are demanding a sum of around €20m for the striker who scored 16 times and provided eleven assists in the Belgian top flight last season.
