Press Association Sport understands Faulkner has stepped down over the club's handling of Stuart Pearce's departure as manager, and the subsequent appointment of Dougie Freedman as his successor.
The former Aston Villa CEO was brought in towards the end of August to help stabilise Forest's infrastructure following the damaging reign of ex-boss Billy Davies and was working to get the club back in line with Financial Fair Play (FFP).
Faulkner spent more than three years in the same role at Villa Park, building a solid reputation for himself
But his influence at the City Ground was not the same and it is understood he was not involved in the appointment of Freedman.
In fact, Faulkner had advised Reds owner Fawaz Al-Hasawi to be patient with Pearce but the Kuwaiti decided to sack the former Forest captain on Sunday following a run of only three wins in 23 games.
Freedman becomes Al-Hasawi's seventh manager in the space of a little more than two-and-a-half years.
Forest are 12th in the Sky Bet Championship and 13 points adrift of the play-off places.
Because of the transfer embargo placed on the club for breaching FFP rules, they are not allowed to sign players but have space in their squad for one more loan signing provided there is no loan fee attached to the deal.
Source : PA