The Istanbul giants have been on the lookout for young Turkish players and are preparing to make moves in the transfer market in January.
The 22-year-old forward has been singled out by manager İsmail Kartal as a top priority following his good performances in the Super League this season.
Şahiner scored the winner against Karabukspor two weeks ago and scored against Trabzonspor in his sides 3-2 defeat to the Black Sea Storm.
Fenerbahce manage Kartal previously worked with Şahiner while at Konya Şekerspor and has been monitoring his progress ever since. Şahiner earns €1.75 million per season and has has a contract with Konyaspor until the end of the 2016-17 season.
Şahiner joined Konya Şekerspor at the age of 16 and moved to Konyaspor in 2008. The young striker has six U21 national team caps to his name.
The versatile forward has scored three goals and provided two assists in nine games for Konyaspor this season.