It is claimed Milan are not desperate to add another midfielder, but they are intrigued by Dembele's declining importance to Tottenham and his contract situation.
Dembele has 18 months left to run on his Tottenham deal, and they would like to bring him in, although there is no elaboration whether they could seek to move in January or simply monitor his situation until the summer.
The Belgian would be unlikely to want to quit Spurs mid-season, especially with a Champions League last 16 tie against Juventus on the horizon.
But the midfielder has been a big disappointment this season. He has struggled with a nagging foot injury, and when he has played he has been underwhelming, not the force of campaigns gone by.
If this continues to next summer and Milan come up with a reasonable bid when he has just one year remaining on his contract, Tottenham Hotspur could have a tough decision to make.