The Daily Telegraph linked Enrique with a move away from the club to Crystal Palace on loan. There are a couple of scenarios that may ensure he says goodbye in the near future, whether that is on loan this month, or permanent in the future.
Enrique briefly worked under Pardew at Newcastle United in the 2010/11 season before moving to Anfield in the summer of 2011. He was the regular left-back under Kenny Dalglish and during the first season of Brendan Rodgers' tenure on Merseyside.
However, the combination of a knee injury, the signing and form of Alberto Moreno combined with the struggle of trying to accommodate the 28-year-old Spaniard into the team on a more regular basis means that his first team opportunities have been limited at Anfield this term.
Liverpool may not want to let Enrique go right now because he is cover for the other options in the Liverpool team. But as well as Alberto Moreno playing ahead of him in the left wing-back role, Lazar Markovic has also been getting some game time there of late.
It does make one wonder whether the more senior left-back is really in Brendan Rodgers's plans, and it will be even more interesting to see how his position in the side is affected when Jon Flanagan returns to full fitness. Whether that happens this month or in the future remains to be seen.
But if Flanagan is given more opportunities in the side on the left flank between now and the end of the season with Enrique still at the club having recovered from a similar injury the Spaniard suffered, then it's hard not to think that the writing is on the wall for the former Newcastle United man.
As mentioned in the Telegraph's report, this is a busy period of fixtures coming up for Liverpool with the Premier League, a two-legged League Cup semi-final against Chelsea and an FA Cup tie against Bolton Wanderers coming up.
Brendan Rodgers may look to keep his side as fresh as possible going forward, so it wouldn't be wrong of Enrique to expect to start at least one of those games - with the clash against the Trotters at Anfield the most likely one. But failure to play much part in any of those matches would suggest there is little faith in Enrique from his manager during the busy periods.
If that is the case, it could be for the best that he finds a new challenge.