The England international has been absent for the Gunners' campaign this season following an ankle injury picked up in the summer.
"It's coming along well, I'm working hard in the gym every day and feel I'm in good hands at Arsenal," said Wilshere.
"We've had a few injured at Arsenal in recent years, like Aaron Ramsey, but they are helping me through it. I can't put a specific date on it, but some time in January I'm hoping for if all goes well."
The 19-year-old underwent surgery in September after suffering a setback in his recovery.