The 28-year-old struggled for playing time in his second season at Atletico, featuring in 18 matches in all competitions in 2019-2020.
Leverkusen, which has sold Kai Havertz and Kevin Volland to Chelsea and Monaco respectively this transfer window, must reportedly pay €12million to sign Arias permanently.
"Bayer 04 Leverkusen are one of the best clubs in the Bundesliga with a very good reputation in Spain. I saw for myself how football is played here with Atlético in the Champions League, that's why I'm really looking forward to the German league and meeting my new teammates." pic.twitter.com/AyBoKRfAH8— Bayer 04 Leverkusen (@bayer04_en) September 24, 2020
He becomes the German club's second signing of the window following the addition of Patrick Schick from Roma on a permanent deal earlier this month.
Leverkusen confirmed Arias's arrival and the club's sporting director Rudi Voller said: "We have signed a very experienced player who can quickly get up to speed to help us.
"Given the high amount of games in the coming weeks, it was important for us to strengthen our team."
Colombia international Arias, who made 51 appearances for Atletico in all competitions after joining from PSV in 2018, is relishing his new challenge in the Bundesliga.
"This is a great opportunity after two great and successful years in Madrid," he said. "Leverkusen are one of the best clubs in the Bundesliga with a very good reputation in Spain.
"With Atletico in the Champions League, I saw for myself how football is played here."
Arias had also been linked with Roma and Everton. His departure from Atleti follows Alvaro Morata's move to Juventus, while Luis Suarez was signed from Barcelona on Thursday (AEST).