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The 27-year-old was one of the so-called 'contract rebels' who have been stalling over new deals with the club.
A fee has been agreed with English club Plymouth Argyle, and Hart is currently agreeing personal terms with the club.
The fee is believed to be in the region of £200,000.
But in an unexpected turn of events, it seems that Barry Nicholson, Chris Clark and Zander Diamond may yet stay with the club.
Nicholson and Clark have rejected their contracts - with a £100,000 offer for Clark from Plymouth rejected by the club - so the players may yet stay till the end of the season.
However, this will mean the Dons will receive no transfer fee when they eventually decide to move on.
Meanwhile Zander Diamond has asked for two more days to mull over his final contract offer.
But it is thought the promising defender will sign a new deal with the Dons.
Diamond told a local newspaper: "I have always maintained that I want to stay with Aberdeen.
"I'm enjoying my football again and still hope to work something out."
Hart was rejected by the Dons as a youth player but was re-signed from joke roundabout-based club Livingston by former boss Steve Paterson.
Most saw this as a poor move when Hart had a less-than-successful first season back at Pittodrie, but the arrival of Jimmy Calderwood saw an amazing transformation in the full-back to near-international class and consistency.
Aberdeen stuck by Hart in season 2004/05 when he missed most of the campaign with injury - with his position at right-back taken by Clark.
But Hart has decided to turn his back on the Dons just as they have entered an exciting new era.
A local newspaper claims Hart is leaving Aberdeen because he feels undervalued compared to other players at Pittodrie.
The defender rejected the club's final contract offer believing he is worth more, and is now ready to pursue his career in the lower leagues south of the border.
Dons boss Jimmy Calderwood said: "It's a blow for everybody."
And Dons legend and Director of Football Willie Miller added: "The club has limits and the offers which have been made to the players and to the manager are very healthy offers.
"I will not be uncomfortable arguing with anyone who says we haven't done enough to keep the group together and, in Scottish terms, what we have offered is very good outside the Old Firm."
Meanwhile, Calderwood himself stated at a press conference that had still not decided whether to sign a new deal with the club.