The 23-year-old, who has started for Jacksonville in the last 24 games, was placed on injured reserve after damaging his right forearm in the overtime defeat to the Houston Texans on Sunday.
Gabbert has also been troubled by a torn labrum in his non-throwing, left shoulder since late last month and will have surgery to repair that on Monday.
"It's something you don't want to have to deal with but, at the same time, it's a risk that is part of the game," said Gabbert, a first-round pick in the 2011 draft.
"I'm hoping for the best and (that it's) something that can be corrected and healed up fairly quickly.
"The biggest thing for me is I've got to get healthy. I've been playing through the shoulder for five weeks now and with the arm keeping me out, I have to get that fixed."
Henne had already been named at starting quarterback for this weekend's game against the Tennessee Titans, when the Jaguars will be out to improve on a dismal 1-9 record this season.
"Hopefully, I can keep it going from last week, and give this offense a spark we've been missing," said Henne, who completed 16 of 33 passes for 354 yards and four touchdowns after taking over from Gabbert against the Texans.
"I'm really excited with what we're doing. We'll see what happens. This game isn't just about one position, but the better the quarterback plays, the more points and the better the offense plays."
Henne, 27, played for the Miami Dolphins from 2008-2011, starting in 31 games, before signing a two-year deal with the Jaguars in March.
