Bilic was approached about becoming manager before Hodgson began his reign in 2007 - and Fulham will try to tempt him away from his national side again as they move into a new era.
The 41-year-old remained in contact with Fayed during Hodgson's time at Craven Cottage.
The Fulham owner even invited the whole Croatia side to Harrods when they played England in September, and is hugely impressed by Bilic's list of European contacts and impressive knowledge of potential transfer targets overseas.