Hart, who had angered Livingston by signing a pre-contract agreement to return to the Dons in the summer - will now move within the next 24 hours.
The fee is believed to be in the region of £30,000.
Hart could make his debut on Tuesday night, when Hearts come to Pittodrie as the SPL resumes after the winter break.
If the Dons can beat Queen of the South on Saturday - Hart will be eligible for the Scottish Cup 4th Round.