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Hart, 27, will move to Deepdale when his contract ends in the summer unless compensation can be agreed with Preston during the January transfer window.
The move will be a blow to the Dons as they had accepted a near-£200,000 offer for Hart from Plymouth last week.
But the player could not agree terms and has instead opted to try and help Preston avoid relegation to the third tier of English football.
This could mean that Aberdeen receive no fee at all for one of their best and most consistent players over the last few seasons.
Preston North End Chairman Derek Shaw told the club's official website: "We made Aberdeen an offer earlier this month which was not accepted. I understand that Aberdeen have agreed a fee for two of their players with another Championship club, however Michael has chosen to come to Preston and we are delighted with that.
"Ideally we would like to agree compensation with Aberdeen in the January transfer window and Michael can start his Preston North End career sooner rather than later, however if that doesn't happen then Michael will be our player in time for pre-season in the 2008/09 campaign."