The Toffees are keen to add Walsh, who is also a coach for the Premier League champions, to their set up with still over a month left of the transfer window.
Walsh played a key part in the Foxes’ title winning season, telling the club to sign Jamie Vardy, Riyad Mahrez and N’Golo Kante. All three featured in last season’s PFA Team of the Year.
Everton want Walsh to become the Director of Football at Goodison Park, in a summer of change that has seen Ronald Koeman leave Southampton to take over the vacant manager position at the Merseyside club.
Sky Sports believe Walsh is keen on discussing the idea of moving to Everton, despite the Foxes reportedly turning down the offer. Neither club have given an official statement on the offer.