He has spent the past 14 months out of management and wants another crack at it.
The news is reported by Sky Sports.
Hughton is the favorite for the job, and has already turned down five jobs in the Championship in the past year.
Another one who is on the list of possible candidates is Mick McCarthy, who left the Republic of Ireland job last week.
Bristol are currently 12th in the Championship and managed by caretaker Dean Holden.