He has not played since Gloucester's Guinness Premiership defeat at the Rec against Bath in October after damaging a disc in his neck.
Brown has missed seven competitive matches in all competitions but his return would be a major boost to head coach Dean Ryan, who will look to protect Gloucester's 100% European record this season.
"Alex will be back in full training this week and we will see how he reacts. If he does ok, he could be available for the weekend, which would obviously be a major advantage to us," Ryan told www.gloucesterrugby.co.uk.
"He is an exceptional line-out forward and any team would miss a player of that calibre. We will just monitor how he progresses in a full week of training."
Gloucester's medical team will also closely watch flanker Andy Hazell, who damaged a knee in the 21-12 Powergen Cup defeat at Bath last weekend.
He has a sprain and Ryan added: "It's a case of monitoring Andy this week to see how he reacts to treatment. At the moment he is not ruled out of the game this weekend."
Meanwhile, James Simpson-Daniel, who damaged a hip during England's victory against Samoa nearly 10 days ago, ran in training yesterday and is also making good progress.
The KSM had scheduled to hold its annual Christmas Quiz at Kingsholm on Tuesday, December 6 but that date has been cancelled, since Gloucester Rugby asked KSM to participate, with the club, in the O2 Premiership Clubs Rugby Quiz scheduled for 13 December at the same location.