Real are reported to be seeking €25million for the midfielder, €7million less than they paid for him 18 months ago.
Illaramendi started Saturday’s 2-0 win over Deportivo La Coruna and The Rossoneri were linked with a move for the former Real Sociedad player in January.
However the move never went through amid reported interest from Athletic Bilbao.
‘Illara’ is held in high regard at the Bernabeu and holds the distinction of commanding the highest fee the European Champions have ever paid for a Spanish player.
But his position has come under threat recently with the signing of Cruzeiro midfielder Lucas Silva and reports that Casemiro may be recalled from FC Porto when his loan there expires in the summer.
And with Luka Modric edging back to fitness, it seems Pippo Inzaghi’s side would be a viable option if Illaramendi’s playing time was reduced.