The 26-year-old joined Arsenal from Real Madrid for a club record fee of £42.5 million back in 2013.
“Fenerbahçe are serious but it all depends on what answer Arsenal give. It will all become clear at the end of the season,” Söğüt told Vatan.
Özil has previously stated that he would like to play for the team he supports one day.
Özil was born in Germany and is part of the largest Turkish community based in the country with an estimated population of 3-4 million.
The World Cup winner often travels to Turkey to meet family and is fluent in Turkish but the £42.5 million record Arsenal transfer still has another three-years left on his contract and is reported to be happy at the club.
The North London based club take on reigning Premier League champions Chelsea on the weekend at Stamford Bridge – Özil is expected to start.