The Brazilian was forced off the field during the Reds’ 2-0 win away to Huddersfield Town, replaced by academy goalkeeper Shamal George.
Having featured heavily throughout pre-season so far, Lucas underwent a scan that revealed frustrating results.
Lucas will also be unavailable against Chelsea, AC Milan, AS Roma, Barcelona and Mainz during pre-season.
Heavily linked with a move away from Merseyside this summer, with Turkish giants Galatasaray said to be weighing up a permanent move for his services, injury could put paid to Lucas’ plans.
The 29-year-old has endured a series of injury problems during his time with Liverpool, with an ACL rupture suffered in 2011 affecting his mobility on the field.
Though a tough-tackling midfield presence, this has seen Lucas jar with Klopp’s tactical ideals, with Germany international Emre Can preferred as the manager’s first-choice holding player.
Grujic has scored a goal in each of his pre-season appearances so far this summer, having joined Klopp’s squad after six months on loan with Red Star Belgrade.
Klopp has also added to his midfield ranks with the signing of Georginio Wijnaldum from Newcastle United, with the £25 million star able to operate in a number of roles in the middle of the park.