That rules him out of the League Cup showdown with St Mirren on February 2 and would make him doubtful for the final on March 21.
"We have a very tough semi-final coming up against St Mirren and that's going to be a difficult match," Thompson told his club's official website. "It's really disappointing to miss that but if we can get through then my aim is to do all I can to be available just as soon as possible, without taking any risks.
"I know it's early days and it depends on how the rehab goes but we reckon about eight weeks, so hopefully I can be available for selection at what is always a hugely important part of the season."
Thompson has impressed since making his first-team debut against Dinamo Zagreb in the Europa League in August, and he is determined to return an even better player.
"I was injured in the game against Celtic at Tynecastle so it's going to be a while that I will have been out of the team," he added.
"But hopefully I can come back better and stronger.
"I had the op on Saturday and have been given a rehab programme to follow.
"It is something that had to be done but hopefully it is not going to be too long before I'm back into things."
Rangers 1/3, Draw 4/1, Hearts 8/1
Source: Team Talk
Source: Team Talk