The Frenchman has been without a club since being sacked by West Brom last month following his allegedly anti-Semitic 'quenelle' gesture.
Atletico president Alexandre Kalil tweeted on Sunday morning: "Anelka e do Galo" - translated as "Anelka is a rooster" - in reference to the club's nickname.
The Copa Libertadores champions, whose roster includes Brazil star Ronaldinho, have yet to confirm the signing.
Anelka was fined by a Football Association-appointed independent panel and handed a five-match ban for the controversial goal celebration he performed at West Ham on December 28.
The former Arsenal and Real Madrid was suspended by his club but, angered by a continued inquiry, announced he would terminate his contract at The Hawthorns, only for the Baggies to fire him hours later.
The infamous pose that Anelka struck after scoring at Upton Park has been associated with religious intolerance in France.
But the 35-year-old insisted throughout the investigation into his actions that it was rather an anti-establishment rather than anti-Semitic gesture, in support of his friend Dieudonne M'bala M'bala.