The Toffees are keen to secure the signing of the 21-year-old both to sure up their midfield options, but also to fend off interest from other Premier League outfits includying; Leicester City and Fulham.
The 21-year-old scored 15 goals last season for the Championship side and was voted Norwich City's player of the year, making him an attractive option for several top-flight teams.
For their part, Norwich will be hoping to cash in on their midfielder, with Maddison's value estimated at between £20-£25million, an impressive return for the Canaries who signed him for £2.5million in 2016.
Maddison is currently recovering form knee ligament damage that he sustained on the last day of the season against Sheffield Wednesday.