The Swedish striker is in talks about a loan move back to Tynecastle Park, where he scored 23 goals in 52 games between 2014 and 2016.
Hearts want Sow to replace Kyle Lafferty after selling him to Rangers and a deal could be struck quickly if all parties agree.
Medical staff at Riccarton will first check on his physical condition before any agreement is signed off.
The 28-year-old is available on loan from the English League Two club and would offer Hearts another attacking option as they wait to learn if David Vanecek will arrive in this transfer window.
Paul Tisdale, the MK Dons manager, said: "Osman wants to look at other options. He is in talks with another club and we are in talks with that club for him to move but it hasn't been concluded yet."
Sow joined Hearts in 2014 after leaving Crystal Palace and quickly became a favourite with supporters by scoring the winning goal against Rangers on his league debut.
He left to join the Chinese club Henan Jianye for a fee of more than Â£1million in 2016, and then played for Emirates FC before moving to MK Dons 12 months ago.