The Blues were first linked with the highly rated England U21 star nine days ago, with Sky Sports claiming they were leading the race for his signature.
Maddison was a star for the Canaries last season, scoring 14 goals and registering eight assists. His form is reported to have caught the eye of several of the top six sides, including Manchester City.
Everton appear to be leading the race, though, with Leicester City their main rivals as far as the Mirror are concerned.
They report that both clubs could make a move as soon as possible as they look to snap him up before the likes of City and Tottenham make their initial interest more concrete.
It is believed Norwich would be looking for between £20-25m for the 21-year-old, who won the club’s Player of the Year award last season and was named in the PFA Team of Year.
They are reluctant to sell but with parachute payments from their relegation in 2016 coming to an end they may be forced to sell their prize asset.
On the face of it the move makes sense for Everton, with Marco Silva and Marcel Brands keen to inject the current squad with young and hungry players of Maddison’s ilk.
The imminent departure of Wayne Rooney and Davy Klaassen doesn't necessarily mean there is a vacancy in a crowded midfield, especially with youngster Kieran Dowell waiting in the wings.
However, with the current group already stocked full of central midfielders it seems likely that some will have to leave before any potential move for Maddison comes to fruition.