Team captain John Muldoon heads that list, after missing the recent Magners League games against Leinster and Edinburgh with a groin injury.
Backs Niva Ta'auso (knee), Keith Matthews (hamstring) and Liam Bibo (calf) and forwards Robbie Morris (calf), George Naoupu (hamstring) and Ray Ofisa (vertigo) have also been sidelined in recent weeks, and could miss Connacht's biggest game of the season.
Commenting on the province's injury situation, Bradley said: "We will wait as long as we can during the week to give them the best chance of being fit.
"It's part and parcel of the game, so we will just have to wait for progress reports. It is a huge game for us and we are all looking forward to it.
"We are going into it on the back of some good form so hopefully the injuries will clear up as a much as possible."
Connacht go into the Bourgoin match boosted by the 22-21 victory over Edinburgh last Friday night, and they will put their seven-match unbeaten run at the Sportsground on the line against the French side.
Bourgoin, winners of the inaugural Challenge Cup in 1997 and runners-up the following year and again last season, went down 25-15 to Brive at the weekend in the French Top 14.
The winners of Saturday's encounter will enjoy home advantage in the semi-finals of the Amlin Challenge Cup.
Tickets are available online through Ticketmaster or by calling in person to the Connacht Branch headquarters at the Sportsground during office hours.
Tickets are available at the following prices: Ground tickets (€25), Concession (€20) and Child (€10). Prices do not include Ticketmaster booking fee.
CONNACHT Squad (v Bourgoin) - Backs (11): Fionn Carr, Gavin Duffy, Ian Keatley, Keith Matthews, Frank Murphy, Troy Nathan, Miah Nikora, Conor O'Loughlin, Niva Ta'auso, Brian Tuohy, Aidan Wynne.
Forwards (15): Andrew Browne, Sean Cronin, Adrian Flavin, Jamie Hagan, Ronan Loughney, Mike McCarthy, Mike McComish, Robbie Morris, John Muldoon, George Naoupu, Dave Nolan, Johnny O'Connor, Michael Swift, Bernie Upton, Brett Wilkinson.