Bristol City sacked Derek McInnes on Saturday following a 4-0 home defeat by Leicester City which sent the club to the bottom of the Championship.
Coventry City boss Mark Robins was mooted as a contender for the position, but it is understood that O'Driscoll is the leading candidate for the role.
O'Driscoll was dismissed by Nottingham Forest on Boxing Day after lasting less than six months in charge at the City Ground and he has already turned down a return to management with Barnsley.
He has also been linked with the Blackpool post following Michael Appleton's departure to Blackburn Rovers, but O'Driscoll is reported to favour the Ashton Gate hot-seat.
Speaking after the sacking of McInnes, City managing director Jon Lansdown told the club's official website: "This is a decision we felt we had to make for the good of the football club.
"Derek has been a pleasure to work with; he's disappointed but he understands the decision.
"We want to give ourselves the best possible chance of getting out of the situation we are currently in and we've got 20 games to do that."