The 52-year-old, who played for Chelsea between 1996 and 2003, returned to the club last summer on a one-year deal to work alongside Maurizio Sarri.
It is believed a role as a club ambassador has been offered to Zola but the Italian would prefer to continue coaching and is unlikely to accept the position.
And with Sarri leaving Chelsea for Juventus on a three-year deal, it is thought Zola may follow him to the Serie A champions.
However, it is believed Zola is keen to stay and look for opportunities in English football.
Derby boss Frank Lampard is the overwhelming favourite to succeed the Italian with another former Chelsea player, Jody Morris, expected to become assistant after working with him at Pride Park.