Manager Arsene Wenger is on the lookout for a new defender following the £7million sale of Gael Clichy to Manchester City.
Wenger has been monitoring Enrique for some time and has high hopes of luring the 25-year-old to north London.
The Spain international is believed to be keen to leave St James' Park but Arsenal face strong competition from Liverpool, who were also in the hunt to sign Clichy.
Meanwhile, retired Manchester United midfielder Paul Scholes has hit out at the Gunners for not winning anything in six years.
Once locked together in a two-way battle for supremacy at the top of the English game, United and Arsenal's rivalry was notorious, with Scholes often at the heart of it.
But while the Gunners have been unable to match United in the trophy department and have not lifted any silverware since their 2005 FA Cup win over the Red Devils.
Scholes: "They just flatter to deceive.
"They may play the prettiest football, but it doesn't always produce the results they need.
"It doesn't irritate me that people say they're the best footballing team because while they are doing that, we (United) are winning games.
"They do play the best football to watch at times, but what is the point of that if you are not winning anything?
"Not only that, they are potentially going to lose their best players in (Cesc) Fabregas, (Samir) Nasri and (Gael) Clichy.
"I don't think our manager would allow this club to go six years without a trophy."
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