Celtic are in talks with Tottenham Hotspur to take Hart to Parkhead this summer in order to solve their goalkeeping crisis.
Vasilis Barkas has continued to struggle between the sticks and Scott Bain was unconvincing against Midtjylland on Wednesday night.
The veteran goalkeeper has emerged as a target and Celtic are trying to sign the former Premier League winning shot-stopper.
But it has been claimed that he is not a name Postecoglou requested from the Celtic board this summer.
It is unclear who the new Celtic boss wanted but Hart is not the goalkeeper he requested for this summer.
However, he is open to signing him as he is desperate to bring in a goalkeeper of some stature to provide some stability to the backline.
Hart is in the final year of his contract and could be keen on the move to Parkhead if a deal can be agreed.
He has played just 13 times in the last two seasons and is keen to get back to playing regular football.