Championship pair Plymouth and Preston and Edinburgh duo Hearts and Hibs are all keen on signing the defender whose contract expires in the summer.
However, Plymouth want their man now and are reported to have had a £200,000 bid accepted for the vice-captain.
Hart joins Barry Nicholson and Chris Clark on the way out at Pittodrie with Zander Diamond's future also still hanging in the balance after Director of Football Willie Miller increased his offer to the player.