Mensah went down during a practice game 24 hours ahead of their Group D meeting with Australia.
Ghana's Serbian coach Milovan Rajevac told reporters both Mensah and fellow-defender Isaac Vorsah, who is also injured, were doubtful for the game.
Rajevac, speaking through an interpreter, said: "We have other players to replace them."
He was unable to explain the injuries but Ghanaian reporters said Mensah may have suffered a repeat of the back-related injuries that he has suffered for two years.
Vorsah's injury was not explained.
Rajevac stressed the importance of the result and said he had warned his players against complacency following their 1-0 victory over Serbia.
"For us the result is all-important and we must be motivated by our opening win because Serbia won today (against Germany) and they are back in it now."
Striker Asamoah Gyan, who scored from the spot against Serbia, said: "In 2006 we did quite well, we went to the next stage but this time Ghanaians are expecting more from us.
"That is why we are also going like wounded lions. Like a wounded lion, you know? So I'm hoping we just qualify for the next stage and then we see what happens from there."
‘Scoring in a World Cup is like winning the title – can you imagine millions of people celebrating something you did? It’s insane and made me very proud’: Brazil legend explains how much 2002 goal meant to him
'He already had the ability but didn’t use it because he was afraid to shoot – I said, "You’re a player who has to decide games – you have to take risks, mate"': Liverpool star was forced to become confident by team-mates
