Enrique picked up the hamstring problem during Liverpool's 3-0 win at Queens Park Rangers last weekend.
The Spaniard was forced to miss the midweek win over Sunderland and he is unlikely to return before February.
Liverpool boss Brendan Rodgers admitted it was a blow to lose the in-form left-back.
"He has looked fantastic in the games but it looks like he'll probably be out for up to six weeks with his hamstring," said Rodgers.
"It's a shame because in the game he felt nothing. Sometimes you can enter the games and be tight and stiff. He felt great, and even within the game."