The influential defender picked up an ankle injury during the north London derby against Tottenham and it looked as though he could miss the busy Christmas schedule.
However, the defender says his recovery is going better than expected and he hopes to be back knocking on Arsene Wenger's first-team door within the next few weeks.
Bellerin told the Daily Mirror: "It is a hard injury to recover from and a bit painful. But I am going to try to be running from this week and hopefully I'll be back on the pitch in less than four weeks."
The game against Manchester City on December 18 will be touch and go but he should be back for the games against West Brom, Crystal Palace and Bournemouth in the space of eight days.
The news comes as a boost after Mathieu Debuchy picked up a hamstring injury at the weekend and is facing a lengthy spell out.