Rodriguez seals Liverpool switch
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 defeat against Reading.
La Liga Loca blog: Will Maxi be born again at Anfield?
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.
"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.
To read what our Madrid-based blogger thinks of Maxi, see La Liga Loca