The 21-year-old England international was taken off after 23 minutes following a challenge with Clarets captain Ben Mee.
Burnley boss Sean Dyche said it was a “fantastic tackle” by Mee.
A Liverpool statement on Thursday said “initial indications” were that Gomez would be out for up to six weeks, “though this is dependent on how his treatment and rehabilitation programmes progress”.
Gomez has played in 13 of Liverpool’s 15 league games this season.
He made his England debut last year and has won six caps, but missed this year’s World Cup in Russia because of an ankle injury.