The goalkeeper has been in fine form this season and his performances have reportedly attracted the interest of the Serie A champions
The San Siro side are searching for a new keeper to compete with Christian Abbiati for the No. 1 jersey, and they consider the 29-year-old to be a valuable addition to their squad.
Demirel started his professional career with Kartalspor before joining Fenerbahce in the summer of 2002. The Turkey international has since developed into a key player at the Istanbul giants and has made well over 150 league appearances during his time there.
His current contract until 2013, but Fener could decide to cash in if a good offer comes in this summer.
Meanwhile, defender Gokhan Gonul has revealed that he could leave Fenerbahce if the club receive a suitable bid for his services in the close season.
"I am ready to move on if a bigger club than Fenerbahce meets the club's financial demands," Gonul said to Lig TV.
The 26-year-old has been linked with La Liga giants Barcelona in recent weeks, and has also caught the attention of a number of Bundesliga sides.