The 20-year-old joined Celtic from Motherwell two years ago but has been troubled by injuries.
Earlier this season he was on the bench for the Europa League home match with Sion but has found his opportunities limited.
MK Dons are currently sitting in fith place in League One and demonstrated their ambitions by taking former England midfielder Alan Smith on-loan from Newcastle United.
The deal was pushed through late last night and is dependent on international clearance.
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