The 34-year-old, who has only featured on 10 occasions for Spurs during the 2020-21 campaign, still has another 12 months left to run on his contract at the Tottenham Hotspur Stadium.
However, according to Italian journalist Fabrizio Romano, Celtic are in talks with the Premier League club over a potential deal in the current market.
The former Manchester City and England number one made 24 appearances during a two-year spell at Burnley before joining Tottenham on a free transfer.
Hart has made 340 Premier League appearances during his career, while he has also featured on 39 occasions in the Champions League.
Celtic will not compete in this season's Champions League, though, having lost 2-1 to FC Midtjylland in the second leg of their second qualifying round contest on Wednesday night. body check tags ::