Rubio, who is under contract through this season, had an MRI on the knee at Cleveland Clinic Sports Health on Wednesday after suffering the injury during Tuesday night's 108-104 loss to the New Orleans Pelicans.
The Cavaliers said they would update his status "as appropriate."
Rubio also tore the ACL in his left knee during his rookie season in Minnesota in 2012.
As Rubio was driving down the lane Tuesday, he maneuvered around Pelicans guard Devonte' Graham and slipped as he planted his right foot, and his left knee buckled as he fell to the ground with 2 minutes, 20 seconds remaining in the fourth quarter.
Rubio, in pain, immediately grabbed his left knee and signaled to the Cavaliers' bench for help. He was not putting any weight on his left leg as he was helped off the floor by teammates Kevin Love and Isaac Okoro.
The 31-year-old Rubio had a major impact in his first season with Cleveland, providing leadership and depth to a young squad. The Cavaliers have already won 20 games after going 22-50 last season.
His injury is the second significant one to Cleveland's backcourt. Collin Sexton suffered a season-ending knee injury Nov. 7. At the time, Sexton was the team's leading scorer.
"Him and I have such a history," Love said after Tuesday's game. "I really care for him. ... We're just completely gutted. He was having such a great season."
In 34 games this season, Rubio, who was acquired during the summer in a trade from Minnesota, was averaging 13.1 points, 6.6 assists and 4.1 rebounds per game.
ESPN's Andrew Lopez and The Associated Press contributed to this report.