Enrique has travelled to the United States with the Magpies squad and they kick off their tour schedule against Sporting Kansas City in the early hours of tomorrow morning.
But with manager Alan Pardew recently admitting the left-back had indicated an unwillingness to sign a new contract at St James' Park, Liverpool are keen to test Newcastle's resolve.
The Reds are still on the look out for a new left-back after losing out to Manchester City in the race for Gael Clichy.
Now Kenny Dalglish, the Liverpool boss, has instructed the money-men at the Merseyside club to work on a deal for Enrique, although Lyon's Aly Cissokho remains a viable alternative.
Dalglish and his scouting team would prefer to sign the Newcastle defender, but a £10m price-tag has been enough to prevent a deal from taking place so far.
Newcastle have been determined to stand firm over the price, but Pardew is getting to the stage where he would like to know which players he has with him going in to the new season.
Enrique, who has rejected an improved extension to a contract that runs out at the end of next season, wants to move on and Pardew would prefer to have the money to reinvest in his squad.
The Newcastle has £8m in his budget set aside for a new striker, with an offer still on the table for Paris St Germain's Mevlut Erdinc.
But for further additions to be made, Pardew knows offloading either Enrique or Joey Barton who is more committed to the cause despite entering the final year of his deal is key to further changes.
Liverpool are aware that Arsenal are in the hunt to bring in a replacement for Clichy and Enrique is also under consideration at the Emirates.
Dalglish, whose first team squad returns to training on Merseyside today after returning from a tour of Malaysia, is keen to make quick progress in his search to find the left-back that will bring an end to his summer spending.
Pardew has also begun to seriously think about a replacement, particularly after he conceded defeat in his pursuit of Neil Taylor.
Taylor's decision to sign a four-year deal at Swansea means Newcastle are no nearer to bringing in a left-back, although they have been linked with a move for Wayne Bridge.
The need for a new full-back has been further heightened by news that Shane Ferguson could miss the start of the season with an ankle problem.
Ferguson had a further scan yesterday on the problem sustained in Friday's friendly with Darlington.
Newcastle's transfer movements have gone quiet in the last fortnight after a blistering start to the summer in which they signed Demba Ba, Sylvain Marveaux, Yohan Cabaye and Mehdi Abeid.
Pardew has admitted his frustration at Cabaye's absence from the club's tour of the United States.
The 25-year-old was the third Newcastle player, along with Barton and Nile Ranger, to be refused entry to the States over visa problems. It was said to be for a minor technical issue rather than a criminal record.
Now Cabaye will join Barton and Ranger in the development squad, which heads for Holland today.
Pardew, speaking in Kansas, said: We have already had word that he will eventually get a visa, but it doesn't help us, because we need it now. It's a three-week process.
It's one of those situations, but for us, it became very frustrating. There was no talking them around, and unfortunately the process was too long for us to rectify.
Mike Williamson, whose wife is close to giving birth, and for sale duo Wayne Routledge and James Perch are also expected to travel to Holland.
But it has emerged Routledge could yet be on his way to Swansea. The Premier League newcomers are in discussions with Newcastle over how a £1.8m fee should be paid and then they are hopeful of reaching an agreement over personal terms.