The defender's season has been blighted by the problem, which is thought to have been caused by his inability to train properly due to his notoriously-fragile knees.
Manager Harry Redknapp said: "He's off to see a specialist in Copenhagen to get a decision on what needs doing next, as to whether he feels he can carry on strengthening and trying to get through without having surgery or whether he needs surgery."
Redknapp has previously expressed his fears that King, 30, would pick up more and more muscle strains due to his lack of training.
He added: "The knee can't improve; there's nothing you can do with the knee.
"That's how it's going to be for the rest of his career."
King has played just seven times this season and his current spell on the sidelines is approaching six weeks.
Meanwhile, midfielder Jermaine Jenas was set for a scan on the calf injury that forced him from the field in Wednesday night's Champions League win over Werder Bremen.
Source: PA
Source: PA