Enrique was recently ruled out for around 10 weeks after undergoing surgery on his knee.
Rodgers is now keen to boost his defensive ranks at Anfield in the New Year.
"It is certainly something that – if Jose is going to be out for a long time – we would have to look at for sure," he is quoted as saying by Goal.com.
"We think Jose will be a few months. It is not a long-term one but will be the New Year for sure.
"It is not ideal; you would prefer two in there to really be challenging. Jose is clearly one of the best left-backs in the league and to have him out is a blow."
Liverpool return to Premier League action on Sunday when they make the trip to Hull City.