The German is into the final six months of his £350,000-a-week contract, but could leave Arsenal this month.
MLS side DC United and Fenerbahce are interested in Ozil, who is yet to make a final decision over his future.
The 32-year-old attacking midfielder, however, has teased that he could well be on his way to Fenerbahce by heaping praise on them.
“I grew up as a Fenerbahce fan as a kid in Germany,” wrote Ozil on Twitter during a Q and A on Monday evening.
“Every German-Turkish person supports a Turkish team when they grow up in Germany. And mine was Fenerbahce. Fenerbahce is like Real Madrid in Spain. The biggest club in the country.”
Fenerbahce president Ali Koc told Turkish media outlet Sporx earlier on Monday: “Mesut Ozil is still a dream.
“Not a single person from Fenerbahce has made a statement. There are issues between Mesut and Arsenal.
“It is a subject we are closer to than in the past. There is nothing ending.”