The 32-year-old striker has fallen out of favour with Diego Simeone at Atletico Madrid and wants to leave the club when the transfer window opens in just a few days.
He is out of contract in the summer but the striker wants to leave the club early and has been trying to engineer a move away from the club in the winter.
Arsenal are keen on Costa, according to Spanish daily AS, and they could have an edge in the chase to land him.
It is claimed that Atletico Madrid do not want to let Costa join a team they could face this season and, with Arsenal in the Europa League, they will not play Los Rojiblancos.
The Gunners have toiled to score goals this season with Pierre-Emerick Aubameyang struggling to find form and Mikel Arteta is believed to be keen on signing a striker.
Costa could welcome a move back to London with Arsenal, but he is sure to have a host of options to choose from.
The striker proved himself in England at Chelsea where he played a major role in the Blues winning two Premier League titles.
Arsenal are keen and are prepared to sign him if Atletico Madrid are prepared to let him go on a free transfer next month.