The 30-year-old Spaniard undergoes the procedure after colliding with Shinji Kagawa during his side's 2-1 defeat at Manchester United and may be fit to play within two weeks
The 30-year-old rushed out to prevent Shinji Kagawa from scoring in the first half but Andre Wisdom's block on the Japan international meant that the midfielder crashed into the Spaniard and left him injured.
Reina was able to continue for the remainder of the game but missed his side's 5-0 victory over Norwich City on Saturday and boss Brendan Rodgers confirmed that he had also aggravated a thigh injury during the loss at Old Trafford.
"Pepe was injured and Brad Jones came in and kept a clean sheet [against Norwich]," the Reds manager told reporters.
"He had a thigh problem before the game last week and he played and the assessment afterwards was it had got worse - plus he had broken his nose.
"He had an operation to fix the nose [on Friday] so we will see how he is by next week and review it from there."
Reina has struggled for form this season and has been put under pressure by backup keeper Jones, who came into the side and impressed during the Spaniard's absence through injury in October.
Liverpool stand seventh in the Premier League after finding form despite a troubled start to the season, six points behind fourth-placed Tottenham with 34 points from 23 games so far.