The 24-year-old has been in impressive form for the Nerazzurri this season, scoring 18 goals in 20 Serie A games.
Los Blancos, meanwhile, are currently 19 points off the top of the La Liga table, in contrast to just 12 months ago when they had notched up a 40-game unbeaten run in all competitions.
According to Italian newspaper Corriere dello Sport, the European champions see Icardi as the solution to their attacking woes and are willing to activate the Argentine's €110m (£98m) buyout clause.
The report goes on to claim that Madrid want to seal the deal during the January transfer window, while Inter are hopeful of improving Icardi's contract and upping his buyout clause to €200m (£178m).
Icardi has made 168 appearances for Inter since his 2013 move from Sampdoria, scoring 96 goals and registering 23 assists.