The British Lions centre has been suffering from the hamstring tear he picked up while on Heineken Cup duty for Leinster last month, but he has responded well to treatment.
Ireland coach Eddie O'Sullivan is desperate to include one of the world's finest players in his starting line-up at the Stade de France and will give O'Driscoll every opportunity to make the game.
O'Sullivan said: "We always knew that Brian's case was going to go down to the wire. We want to give him every chance to play but we won't take risks.
"The decision will be made in the middle of next week at the earliest and I'm happy to leave it that late."
Llanelli back row Simon Easterby and Munster prop Marcus Horan have shaken off their injuries and are named in the 22, but Leinster flanker Eric Miller is still in the treatment room and is ruled out.
O'Sullivan started this week's five-day training session with a 34-strong squad but that has been whittled down to 22, while the side to face France in Paris will be announced next Tuesday.
Forwards: S Byrne (Leinster Blackrock College), R Corrigan (Leinster Greystones), V Costello (Leinster St Mary's College), S Easterby (Llanelli), A Foley (Munster Shannon), K Gleeson (Leinster St Mary's College), J Hayes (Munster Shannon), M Horan (Munster Shannon), D O'Callaghan (Munster Cork Constitution), P O'Connell (Munster Young Munster), M O'Kelly (Leinster St Mary's College), F Sheahan (Munster Cork Constitution).
Backs: G Darcy (Leinster Lansdowne), G Dempsey (Leinster Terenure College), G Easterby (Rotherham), S Horgan (Leinster Lansdowne), T Howe (Ulster Dungannon), D Humphreys (Ulster Dungannon), K Maggs (Bath), B O'Driscoll (Leinster Blackrock College, captain), R O'Gara (Munster Cork Constitution), P Stringer (Munster Shannon).