The deal is now subject to him gaining international clearance but he was at Anfield on Wednesday to watch Liverpool's FA Cup third-round replay against Reading.
Rodriguez, 29, will link up with his former Atletico team-mate Fernando Torres in the Liverpool attack. He also joins fellow Argentine international Javier Mascherano.
"We were looking for players with character and a good mentality," Liverpool manager Rafael Benitez told the club's website (www.liverpooltv.com). "He has played for Argentina and Atletico Madrid and is a player with personality.
"He can play in three positions - on the right, left or as a second striker - and is someone with a very positive mentality, which is what we are looking for."
Liverpool have had a disappointing season, lying seventh in the Premier League table and failing to reach the Champions League knockout stages.