The Telegraph and the Liverpool Echo claim that the club want the highly-regarded talent-spotter to fulfill the new role, as was rumoured earlier to be going to someone like Ajax's Marc Overmars or Sevilla's Monchi.
Speculation that the latter might join Unai Emery at PSG ended this week after the French side opted for another candidate and the former goalkeeper is expected to stay at Sevilla.
Walsh, also Leicester's assistant manager, is credited with finding the likes of Riyad Mahrez, Jamie Vardy and N'Golo Kante, three cornerstones of their stunning Premier League title success last season.
The Telegraph now say that Leicester are resigned to losing Walsh to Goodison this summer, with Ronald Koeman not wanting "day-to-day involvement in the scouting process".
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