The 35-year-old has been without a club since being sacked by West Brom last month following his use of the allegedly anti-Semitic 'quenelle' gesture when celebrating a goal.
Atletico president Alexandre Kalil posted on Twitter on Sunday morning "Anelka e do Galo", which translates as "Anelka is a rooster", in reference to the club's nickname.
The Copa Libertadores champions have yet to confirm the signing.
Anelka was fined by a Football Association-appointed independent panel and handed a five-match ban for the 'quenelle' goal celebration he performed after scoring for the Baggies at West Ham on December 28, leading to him being suspended by his club.
The former Arsenal and Real Madrid man then said on March 14 that he would terminate his contract at West Brom, only for the club to fire him hours later.
The infamous pose that Anelka struck after scoring at Upton Park has been associated with religious intolerance in France, although the player insisted throughout the investigation into his actions that it was merely a show of support for his friend Dieudonne M'bala M'bala.