The player himself said "it would be an hour to represent" the famous club, while Neil Lennon confirmed their desire to sign the Republic of Ireland youth international.
The deal could be delayed, however, due to the complexities of Connell's contract situation.
Celtic are hoping to recruit the midfielder for a £250,000 cross-border compensation fee. In order for them to do so, Connell's contract will have to be terminated as he's a third-year scholar and won't be allowed to move freely.
The player has sought to terminate his deal with the financially-stricken club. Although, according to the latest reports down south, his 14-day notice has been rejected by Bolton's administrators.
It would seem Celtic will have to negotiate a fee for his release.