The 23-year-old is due to have the operation on Thursday, with reports in Spain suggesting he could be out for between two and four months if his recovery progresses as expected.
Asencio has struggled with the injury since the start of the season and was left out of Real Madrid's matchday squad for their opening five games.
He made 34 appearances for the club last season, although his role was reduced during the final three months of the campaign.
Real Madrid have added Ibrahima Konate to their defensive options this season, while Dean Huijsen has also featured regularly alongside him. Antonio Rudiger remains another experienced option in central defence.
Eder Militao is still recovering from hamstring surgery and has been sidelined since April, leaving coach Jose Mourinho with further defensive concerns.
Real Madrid's next La Liga fixture is at home to Villarreal on Saturday, October 10, at the Santiago Bernabeu.