The Blues are allegedly keen on a move for the Leicester City star.
The Blues are expected to be busy this summer as manager Ronald Koeman looks to spend the big transfer budget offered by Farhad Moshiri.
One area the Toffees look destined to improve is the goalkeeping slot, with Tim Howard having departed for America.
Joel Robles is the current number one and they have also signed Maarten Stekelenburg, but neither is viewed as a long-term candidate for the position.
As a result Koeman will look for further additions and one man Everton have been linked to is Leicester's Schmeichel.
He is currently in discussions over a new contract but may be tempted by Everton’s new project under Koeman according to the report.
Claudio Ranieri’s side have also seemingly planned for his departure too by bringing in Ron-Robert Zieler early in this summer transfer window, suggesting Everton may be able to tempt them into a sale.