Bradley said there's concern about further damage to Gabbert's hand.
"It's got to heal and we need to keep it clean," Bradley said. "The big concern would be with all the stitches that it could reopen and then the chance of infection occurs.
"Some of the skin isn't there. It ripped off part of the skin, so some of those [stitches] are even open. The sweat from practice, anything that can get in there [raises] the threat of infection."
Bradley said Gabbert is taking antibiotics and will be evaluated after the Raiders game to see if the injury has healed enough for him to play against Seattle on Sept. 22.
Gabbert suffered a fractured right thumb in the Jaguars' second preseason game, yet Bradley named him the starter over Henne two days later. Gabbert didn't throw a football until the beginning of last week but improved enough to get the call against the Chiefs.
Gabbert, who missed the final six games of the 2012 season because of a forearm injury, suffered the cut during the Jaguars' final possession when he hit his hand on a facemask. He finished 16-of-35 passing for 121 yards and two interceptions, including one that was returned for a touchdown.
Henne threw for 2,084 yards, 11 touchdowns and 11 interceptions in place of Gabbert last season. Ricky Stanzi, acquired by the Jaguars on Aug. 27 off waivers from the Chiefs, will be the backup against Oakland.
Bradley said he'll talk with general manager Dave Caldwell about possibly adding a veteran quarterback this week.