Marco Silva's side are keen to hold onto their prized asset as they aim to do more business themselves in the final two weeks of the transfer window, having already landed Richarlison from Watford and waved goodbye to Davy Klaassen and Ramiro Funes Mori.
Everton's relationship with Chelsea is strained due to the London side's previous approaches for John Stones and Ross Barkley having proved disruptive for the Merseyside club.
The Blues are said to have made the shot-stopper a top target while they run the risk of losing current No.1 Thibaut Courtois to Real Madrid, as he pushes for a move to the European champions to be closer to his family in the Spanish capital.
Talks for the 26-year-old have stalled as Chelsea want a replacement to come in before sanctioning the sale of their reliable first choice.
However, Pickford is unlikely to be the one to step in for the Belgian, as he only joined the Toffees last year in a £30 million move from Sunderland and still has four years remaining on his contract, but his impressive displays at the World Cup have only increased talk of him heading to a bigger Premier League side.
Chelsea have also been linked with Jack Butland, Kasper Schmeichel and Gianluigi Donnarumma, but a clear option has yet to be determined.
Meanwhile, Chelsea are also in talks with Juventus over the potential signing of Gonzalo Higuain, Daniele Rugani and Mattia Caldara.