Chadwick, part of the squad that won the treble for the Red Devils in 1999, suffered a knee injury on his debut against Ipswich in November. He had made a promising start, getting on the scoresheet in his first appearance.
Canaries boss Peter Grant is now expecting his first signing to provide some much-needed competition for midfield places.
"We are hoping that he can be back in full training on Monday," said Grant. "He's been on the pitch and had fitness tests. We have to make sure he is ready as he's been out for a while and there is a risk of getting injured again."