Sport-Bild in Germany claim the 24-year-old is being watched by Borussia Dortmund, Borussia Monchengladbach and Bayer Leverkusen as well as the two Premier League clubs.
Gregoritsch has had his best season to date with 13 goals for Augsburg, of which eight came away from home.
They paid just 5m euros to sign him from Hamburg last summer and his value has now rocketed into the £10million bracket.
Usually an attacking midfielder, he has been deployed as a second striker and could be an incredible foil to Jamie Vardy if the Foxes decide to push hard for his signature.
Bayer Leverkusen have signed keeper Lukas Hradecky from Eintracht Frankfurt ahead of Bernd Leno's likely exit.