Costa is available as free agent after requesting to end his contract with in Madrid, at the start of 2021, amid rumours of a return to Brazil.
However, the 32-year-old has since failed to secure a new club, with talks over deals with teams in Brazil, Portugal, Turkey and Saudi Arabia all breaking down.
Premier League side Wolves are the latest club to be reportedly interested in a possible deal with new boss Bruno Lage eyeing up the Spanish international ahead of the 2021/22 campaign.
But, according to reports from Turkish outlet Ajansspor, via Marca, Besiktas have now reignited their interest in the former Chelsea man.
The Super Lig club will offer him two year contract including an annual salary of €1.5m plus performance related bonuses.
Costa is now rumoured to be considering their offer alongside other options in both England and China.