Gomez was forced off after 23 minutes following challenge made by Ben Mee in the first half in Liverpool's 3-1 win over Burnley at Turf Moor.
The defender was sent sliding into the advertising hoarding and unable to continue, being replaced by Trent Alexander-Arnold.
Klopp, who voiced frustration with Burnley's tackling following the match, confirmed he expected a serious injury in his post-match conference, telling reporters: “It was his ankle.
“Joe is injured and probably not only a little bit. We have to see, we don’t know in the moment. We sent him home.”
After further assessment on Thursday, the club have determined he will the remainder of 2018 with the fracture, pending his treatment and rehab.
Gomez has become one of the key members of Liverpool's backline this season, having started 12 of the club's 15 Premier League games and four of their five Champions League matches to date.
Primarily acting as Virgil van Dijk's partner at centre-back, the 21-year-old Gomez has also lined up in place of Alexander-Arnold at right back at times this season, which is where he featured before coming off against the Clarets.
Gomez's performances at a club level earned him a place in Gareth Southgate's England side during the UEFA Nations League, starting three of England's four matches as the Three Lions made the semi-finals of the competition.
What matches will Gomez miss?
A six-week timeline would mean Gomez is out the remainder of 2018 – something Klopp will be frustrated by given it mostly coincides with the busy festive period of the Premier League fixture list, which does not let up until the middle of January.
Gomez will certainly miss the Premier League match away against Bournemouth this upcoming weekend, as well as the north-west derby against Manchester United the following Sunday at Anfield.
Sandwiched in-between those games is a midweek clash at home against Napoli, on which all of Liverpool's hopes of advancing to the knockout stages in the Champions League rest, with victory required.
Things do not slow down to start 2019 either, as a trip to Manchester to face the reigning Premier League champions City awaits January 3, followed by an FA Cup match away to Wolves on January 7.
It is possible Gomez may be fit to return for Liverpool's next match, whether that is a potential FA Cup fourth round match on the weekend of January 26-27 or their clash on January 30 against Leicester City, which will have given him nearly a full two months of recovery time.