The 18-year-old was spotted being shown around Celtic Park on Thursday and later agreed to terms on a four-year deal with the Ladbrokes Premiership champions.
The Republic of Ireland youth international joins Neil Lennon's side after leaving his new manager's former club Bolton Wanderers.
There was a dispute as to whether the out-of-contract teenager would be allowed to move for a cross border compensation fee.
Lennon revealed on Thursday that the two clubs were in negotiations and it seems Celtic were able to successfully procure Connell's departure following discussions with Bolton's administrators.
The player becomes the club's second summer signing behind French defender Christopher Jullien.