Hoops boss Ange Postecoglou is in the market for a new goalkeeper and has now agreed a three-year deal with the former England number one, according to the Daily Mail.
Hart has only featured 10 times for Spurs, all in cup ties, since joining the club last summer while his last appearance in the Premier League came for Burnley in a 5-1 defeat to Everton in December 2018.
The arrival of Pierluigi Gollini from Atalanta on loan has limited his first team opportunities at the London club further, with the 34-year-old now keen to revive his career north of the border.
Celtic have had issues in the goalkeeping department stretching back to last season with £5m Greek signing Vasilis Barkas failing to impress and Scott Bain also struggling for form.
Hart has been capped 75 times for England with his last interntional appearance coming in 2017.