The Yellow Canaries had been closely linked with the Nigerian international but will not be pursuing the midfielder further.
“Chelsea want €35 million for Mikel,” Yildirim was quoted as saying by AA at the Topuk Yayla training grounds.
The Istanbul giants are reported to be interested in signing an additional central midfielder following the departure of Emre Belözoğlu.
Fenerbahçe will have made eight signings already this summer following the arrival of Robin van Persei on Sunday.
Mikel played a total of 26 games for the Premier League winning side last season but fell behind Matic in the pecking order and has been linked with a move away from Stamford Bridge.
The Yellow Canaries finished last season in second place and secured a place in the Champions League play-off stages. Fenerbahçe are currently coached by Vitor Pereira who replaced Ismail Kartal as manager last month.