Mullen suffered the injury during last weekend's Edinburgh derby with Hibernian at Easter Road.
It means the 19-year-old will miss this weekend's Scottish Communities League Cup final against St Mirren at Hampden park.
Mullen, who made his Hearts debut in the semi-final win over Invernes, admitted he was devastated to miss Sunday's showpiece event.
"Gutted is an understatement," Mullen told Hearts TV.
"The injury couldn't have come at a worse time but I'll have an operation this week hopefully and then it will be all about getting back fitter and stronger for next season."