The 34-year-old is primed to become the club’s No.1 with Vasilis Barkas and Scott Bain unable to hold down the spot.
Hart, currently with Premier League Tottenham Hotspur, has hinted he is on the way to complete the move to Parkhead.
The 75-time England international shared a picture on social media of him on a plane.
Hart is pictured giving the thumbs up and the post is captioned “Let's go” with a green love heart.
If the deal is concluded in time, the two-time Premier League winner could feature in the Europa League qualifier against Jablonec.
Celtic have until 11pm on Wednesday night to register new signings.
Hart is expected to sign a three-year deal and cost £1million as he seeks first-team football.
In the last two seasons, with Burnley then Spurs, he didn't play a single league game.