The 51-year-old and joint assistant boss Craig Shakespeare only signed new contracts with the Premier League champions in May.
Former PE teacher and Chelsea scout Walsh has been credited with helping discover bargain signings Riyad Mahrez and midfielder N'Golo Kante.
Chelsea paid £30m for Kante last week.
Mahrez joined from Le Havre in 2014 for £400,000 and was a star player as the Foxes became top-flight champions for the first time in their 132-year history.
The Algerian was also voted Professional Footballers' Association Player of the Year last season, but has repeatedly been linked with a move away.
However, striker Jamie Vardy and captain Wes Morgan have signed new deals, with Leicester manager Claudio Ranieri telling Sky Sports on Tuesday that Mahrez "wants to stay".
New Everton owner Farhad Moshiri headhunted "number one target" Ronald Koeman to be the club's manager after Roberto Martinez was sacked before the end of the 2015-16 season.
It has been reported that Walsh, who joined the Foxes in November 2011, could be made Everton's sporting director.
Leicester City assistant manager and head of recruitment Steve Walsh is close to joining Premier League rivals Everton.