Januzaj suffered ankle ligament damage last month in Sunderland's defeat to Crystal Palace.
Januzaj, who is on loan from Manchester United, picked up an ankle injury in the 3-2 defeat to Crystal Palace and was expected to be absent until mid-November, but he has been making good progress after suffering ligament damage.
Moyes is quoted in the Sunderland Echo, saying: “Adnan is feeling much better and he doesn’t think he will be out as long as we initially thought. Hopefully, we can get him back sooner."
The following comment demonstrates the retched injury record that Moyes has had to endure at Sunderland, as well as the poor start to the Premier League campaign.
“We played QPR the other week in the EFL Cup and we had everybody fit, aside from Seb Larsson and Vito Mannone.
“The other players were okay then but suddenly we are into the last couple of games and lost four or five players."
It will be a welcome boost to get the 21-year-old back earlier than expected but it looks unlikely it will be for this weekend's trip to the London Stadium where they take on West Ham United.
Sunderland urgently need to improve on their form having taken two points from their opening eight league games, and they come up against a side that got a morale boosting win against Crystal Palace last weekend.
However, if Januzaj was due to return in mid-November, the chances are he would have been back after the international break for the clash against Hull City.
If he is available beforehand, it would be a boost for Moyes' side to have another attacking option for the trip to Bournemouth.