The Frenchman is under contract with Spurs until 2024.
However, The Mirror reports that the North Londoners want to replace Lloris with Pickford and believe that they can tempt the 28-year-old stopper with the chance of playing European football.
The report adds that the England international would be open to the move, as he continues to refuse contract extensions at Goodison Park, with just 18 months left on his current contract.
The report adds that Chelsea are also keeping close tabs on Pickford as their goalkeeping situation is still unresolved, with Edouard Mendy currently sidelined with an injury and Kepa Arrizabalaga unconvincing.
Lloris has made 439 appearances for the Lilywhites, keeping 149 clean sheets.