The German could be rested against Southampton, Newcastle United or Bournemouth in the coming weeks.
The Gunners have been hit by a number of big injuries over the last month, losing Alexis Sanchez, Francis Coquelin and Santi Cazorla to various problems.
That has threatened to derail their title bid this season, although Arsenal remain second in the table and just two points off surprise leaders Leicester City.
They could ill afford for the German to end up on the side-lines as well, with Ozil leading the assists charts in the Premier League by some distance.
Wenger is therefore thinking of handing Ozil a break over the busy Christmas period to avoid such a nightmare scenario, according to the London Evening Standard.
He is already being monitored closely after reporting tightness in his calf after the win over Olympiakos and was brought off early against Aston Villa as a precautionary move.
With a title clash against Manchester City up next, he is sure to feature in that game but could be rested against Southampton, Bournemouth or Newcastle over the next month.
Given the way the side has lost stars to injury in recent weeks, it is likely Arsenal fans would agree with the move as well.