The England defender, 21, was carried off midway through the first half at Turf Moor following a challenge with Ben Mee, with Liverpool going on to claim a 3-1 win.
"Liverpool can confirm Joe Gomez sustained a fracture to the lower left leg during the 3-1 win at Burnley on Wednesday night," the club said in a statement on Thursday.
"The defender needed to be substituted after 23 minutes at Turf Moor following a challenge from which he was unable to continue.
"Gomez's injury was further assessed by the Reds' medical team at Melwood on Thursday.
"Initial indications suggest the 21-year-old is facing a spell of up to six weeks out of action, though this is dependent on how his treatment and rehabilitation programmes progress."
A six-week absence would keep Gomez out until mid-January, with Liverpool facing nine games between now and then in three different competitions.