Perez, 28, only joined Arsenal in a £17million deal from Spanish side Deportivo La Coruna last summer.
But he has failed to make his mark at the club with just 11 Premier League appearances and one goal to his name.
Sevilla, West Ham, Everton and former club Deportivo have apparently shown an interest in signing the striker.
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Lucas Perez has been offered a £3m-a-season contract by Fenerbahce to leave Arsenal.
But Turkish outlet AMK claim Fenerbahce are in pole position for his services.
It is claimed the the Turkish side are in advanced talks with Perez and have offered him a £3m-a-season deal.
The Turkish side have reportedly made signing a striker a priority this summer as they bid to knock Besiktas off their perch next season.
Moussa Sow looks set to return to parent club Al Ahli following his loan spell and Emmanuel Emenike has been transfer listed.
Ex-Manchester United and Arsenal star Robin van Persie is also on their books, but at 33 he is struggling to produce the fine goalscoring form he showed in the Premier League.
Fenerbahce have offered Lucas Perez a £3m-a-season contract to leave Arsenal
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Perez’s agent has also outlined that the Gunners are willing to sell Perez this summer, but only for the right price.
"We have spoken with Arsenal and we have explained that we want to leave," Lovelle told reporters in Spain.
"They have said that they are not going to put obstacles in the way, but they are not going to give the player away as a gift either."
The report adds the Turkish ginats are also interested in Chelsea striker Michy Batshuayi.
Arsenal have already been busy in the transfer window with Sead Kolasinac officially agreeing to join from Schalke on July 1.