Gomez, who moved across to right-back for the game at Turf Moor, was stretchered off in the 23rd minute after a tackle from Ben Mee that saw him crash in to the advertising hoardings.
Speaking after the game, Klopp confirmed the Englishman was injured and "probably not only a little bit."
Gomez underwent a scan on Thursday and the club have now announced that he suffered a broken left leg that will keep him out of action for six weeks or more.
Klopp's initial fears have been realised, with Gomez missing the entire busy festive schedule, including some crucial games in a season-defining period for the Reds.
Gomez could miss up to nine games for the Reds if he remains out for six weeks. He will miss fixtures against Bournemouth (A), Napoli (H), Man United (H), Wolves (A), Newcastle (H), Arsenal (H), Man City (A), Wolves (A [FA Cup]) and Brighton (A).
His absence will be a huge blow to Liverpool, who have looked accomplished at the back this season with a partnership of Gomez and van Dijk in central defence.
Dejan Lovren will be expected to take his place in the starting eleven, however the Croatian has himself suffered with numerous knocks and has featured just six times this term.
Joel Matip, who made a rare start at Burnley, is the other option for Klopp although the Cameroonian has been limited to mainly just substitute appearances.
Gomez's absence also affects the Reds at right-back, where he rotates with Alexander-Arnold where needed.