The 24-year-old second row cum back row will miss the crucial trip to Perpignan in Round 5 and the final pool game with Munster as a result, with Gloucester desperate for a victory in France that could keep their campaign alive heading into Round 6.
"Tom's had surgery and will be out for at least the next couple of months," director of rugby Nigel Davies told the Gloucester Citizen.
"We have a few injuries as you would expect at this time of year. We're running with just over 20 per cent unavailable and that's why you carry a squad of players and why we had a rotation policy earlier on. That should benefit us because all the players stepping in have had rugby.
"The back row was an area where we went into the season very heavy, arguably over heavy, but as it pans out it's just as well."
Savage missed the back-to-back Heineken Cup encounters with Edinburgh, with his most recent appearance coming in the domestic defeat to Leicester Tigers on November 29.
He has now had surgery on a torn pectoral muscle and joins the likes of veteran wing cum centre James Simpson-Daniel and young playmaker Ryan Mills on the sidelines.
The former Hartpury University captain, who took over the leadership reigns at Kingsholm in the summer, has played 12 times so far this season but will now sit out at least five Aviva Premiership fixtures - starting with the bottom-of-the-table clashes with Worcester Warriors and London Irish this week and next - and two LV= Cup ties, as well as the final rounds of European action in January.
Tongan duo Sione Kalafamoni and Lua Lokotui are also injury concerns for Gloucester, with the former awaiting the results of a scan on a leg injury he suffered on Sunday and the latter currently out with an arm problem.