Neither 36-year-old Azam nor French prop Pierre Capdevielle will be retained next term after reaching the end of their existing contracts at Kingsholm.
Their departures follow those of Dave Attwood, Nicky Robinson, Paul Doran-Jones, Alasdair Dickinson and Jordi Pasqualin, who will all join other Premiership clubs this summer.
Gloucester won the LV= Cup this season and claimed a third-place Premiership finish, losing narrowly to Saracens in the title play-offs.
Head coach Bryan Redpath said: "The current squad has been involved in a memorable season for Gloucester, but it's the nature of the beast in modern rugby that players, some of whom you would desperately love to keep, move on.
"This is never an easy time, as every single player in the squad has played their part in the success that the club has enjoyed this season.
"I would like to place on record my personal thanks for their efforts during their time at the club. They have all been magnificent servants.
"However, we have to move on and prepare for next season.
"We've already announced the signings of Dario Chistolini, Dan Murphy and Nick Runciman, and we are continuing to explore a number of other options and will make announcements at the appropriate time."
Azam has spent 10 seasons with Gloucester, and his current testimonial year is set to include a high-profile friendly against French giants Toulouse at Kingsholm on August 13.
"I am gutted that my services are no longer required at the club, in any capacity," Azam told the Gloucester Citizen newspaper.
"I had always been very loyal to Gloucester and hoped that I would finish my playing career at the club that I love, and then integrate within the staff to really make a difference and to continue my passion for the team.
"I went to great extremes to complete all relevant qualifications in order to make the transition successful for everyone involved.
"Unfortunately, due to opportunity unavailability within the club, now is not the right time for that integration to take place.
"I have no other choice than to look at other options outside of Gloucester, and it is with the deepest of sadness that I do this."
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