The 25-year-old is yet to respond to a new contract offer as he prepares to enter the final year of his existing deal amid speculation that he will head out of St James' Park this summer.
Arsenal were the latest club to be linked with the defender over the weekend with Liverpool having previously been touted as a destination for a player who was one of the Magpies' most dependable performers last season.
However, having lost skipper Kevin Nolan to West Ham after he failed to agree a new deal and with Joey Barton's future up in the air with no fresh offer on the table, Pardew is desperate to retain Enrique's services.
He told BBC Radio Newcastle: "I'm still hopeful that Enrique - who was a big, big, big, big player for us last year - stays at the football club and we can entice him to stay, and obviously we are still working on that.
"Jose's holidaying and doesn't have too much on at the moment, so we are not confident we can keep him, but we are not scared of losing him. It's somewhere in between, really.
"When you have only got a year left on a player of his ability, of course you are vulnerable, and I think it's nervous times for us on him."
Enrique was perhaps always the least likely of the trio to remain in the north-east and many supporters have resigned themselves to his imminent departure.
With Nolan's move to Upton Park coming as something of a shock, Newcastle moved quickly to at least partially appease the dissenters last week when they snapped up Demba Ba and Sylvain Marveaux on free transfers after already recruiting Lille midfielder Yohan Cabaye.
But there is still much work to be done as Pardew attempts to capture his remaining targets.
Swansea full-back Neil Taylor is firmly in his sights, while he has interest in the likes of Tranquillo Barnetta, Blaise Matuidi and Mevlut Erding.
The Magpies have also considered offering Wigan midfielder Charles N'Zogbia a return to St James', although his £12million price-tag and wage demands - understood to be in the region of £60,000 a week - could prove prohibitive.
In addition, few on Tyneside will need reminding that the Frenchman engineered his way out of the club in February 2009 to make the £6million switch to the DW Stadium after being referred to by then manager Joe Kinnear as "Charles Insomnia".
However, the 25-year-old was agitating for a move long before Kinnear even arrived in north-east with Kevin Keegan fighting a losing battle to keep him during his short second spell in charge.
Sunderland manager Steve Bruce was the then Wigan boss who benefited from his defection, and he remains an admirer having twice tried to sign him for the Black Cats, although he would have to argue his case before owner Ellis Short who, like Newcastle counterpart Mike Ashley, is trying to control spending.