The Sun says United boss Alex Ferguson sees the former England centre-half as the answer to United's defensive problems now he is finally allowed to play again.
Campbell, 35, has been training by himself since leaving Notts County in September, just weeks after signing a five-year deal worth £10million with the League Two side.
The one-time Spurs and Arsenal star has now reached agreement with County over his registration which leaves him free to join another club.
United are investigating how quickly they can sign Campbell and the signs are he will be fit for his debut for the champions early in the New Year.
Campbell could face the Gunners on January 31 and another ex-club, Portsmouth, on February 6.