The Magpies’ manager wants the Spanish defender to commit his future to the club before his contract expires next season, but he admits that the chances of that are very slim
The player has been the subject of speculation linking him with a move to other clubs in recent months and has failed to reach agreement over a new contract, with just one year remaining on his current deal.
Pardew told reporters: "He's made it pretty clear that he still won't sign a contract and that his future may be elsewhere.
"I'm not really comfortable with that, but we will have to deal with it and we are still hoping that he signs a contract with us. There has to be doubt over that situation."
Arsenal are believed to be interested in the Spanish full-back as a replacement for Gael Clichy, who left last week to join Manchester City. Liverpool have long been linked with the 25-year-old and it is believed that they had a bid rejected earlier in the transfer window and are still pursuing the player.
Enrique, who has made 119 appearances for Newcastle, signed from Villarreal for £6.5 million in 2007 and was a key player in the club's 2010 promotion season and a consistent presence last year in the top flight.