Pogba, who netted 15 league goals for the Alex last season, sustained a knee injury in a pre-season friendly against a Manchester United XI and has not featured this term.
The 23-year-old was expected to be out for a lengthy period, but has been given the all-clear to start running again and could return ahead of schedule.
Pogba's return would lift a Crewe side that has only managed 12 goals in 14 games, the lowest tally in Sky Bet League One.
"We'll just have to assess how he recovers after running, but it is looking really positive with him and every manager would like to have his better players available as much as possible," Davis told the club's official website.
- Sports & Recreation