Costa is currently in Brazil considering his next move after being turned down by Olympique Marseille last week.
AS says in Turkey, several teams dream of him and the first to make a formal offer was Trabzonspor.
But Costa, who earned €7m-a-year when he was at Atlético, has not responded positively to the Turkish club's proposal, turning it down immediately.
Trabzonspor offered him a clean salary of €2.5m and a three-year contract.
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