The 32-year-old has not played for Arsenal since last March after being frozen out by first-team manager Mikel Arteta.
Agent Sogut is hopeful his client will soon find a new club, with discussions already under way with clubs over a possible deal in the January window.
"Ozil is not an ordinary Fenerbahce supporter. He's a fanatic. He would only come to Turkey to play for Fenerbahce. I hope it will happen," Sogut told beIN SPORTS Turkey.
"This transfer needs a process. It would be wrong to say certain things before something happens, but I hope it will happen."
Sogut added: "It'll be a great moment for Mesut to come to Fenerbahce, as a big fan. But at the moment, I can't say what the chances are. We have to see both sides of the picture.
"Before the papers are signed, nothing is definite. Even if we come to a verbal agreement before contracts are signed, nobody can say this deal is done."
Only Kevin De Bruyne (74), David Silva (63) and Christian Eriksen (62) have provided more Premier League assists than Ozil's 54 since the start of 2013-14.
Arteta hinted last week there is no way back for the German at Arsenal, but Sogut suggested he may remain in north London until his contract expires.
Sogut said: "If this is a good move, let's hope it happens; if not, we don't want it. Let's wait for the process and people now understand what Mesut wants to do from his tweets."
He said: "There have been no mishaps, he trains with the first team and a lack of match time is a problem that can be solved quickly. He'll become a 90-minute player in a short period of time."