Costa terminated his contracted at Atleti on 29 December in the hope of finding a new club and regular first-team football but the transfer window has come and gone, with the striker still without a new team.
A report from the Spanish version of Goal.com has now outlined that the 32-year-old striker is in talks to join Portuguese giants Benfica.
Previously, El Mundo claimed that Costa was only allowed by Los Rojiblancos to depart on a free on the basis that he would not join any rival club.
It is reported that the Lisbon-based club are offering the striker a two-year contract on wages of €3m annually.
Costa had fallen from prominence at Atleti in recent months with Luis Suarez and Joao Felix the established attacking partnership, while Moussa Dembele subsequently arrived on loan from Lyon.