Wenger will depart the Emirates at the end of the season, after a 22-year reign at the club and a number of high-profile names have been linked with the soon-to-be vacant position.
Germany coach Joachim Low, Juventus boss Max Allegri and Gunners legend Patrick Vieira, have all been highlighted as contenders for the job, but it is Enrique who has emerged as the leading candidate.
The Spanish manager has been out of work since leaving Barcelona at the end of last season, and The Daily Mail reports that the 47-year-old has contacted Arsenal to confirm that he is interested in taking over at the Premier League side.
Enrique is understood to have told the London club that he would relish the challenge of replacing Wenger, and the former midfielder is thought to be the favoured choice of head of football relations Raul Sanllehi, who previously worked with Enrique at Barcelona.
Enrique won two La Liga titles, three Copas del Rey and the Champions League, during his three year reign at Camp Nou, guiding Barcelona to the treble in his first season in charge.
The Spaniard is a well respected manager, and Spanish football expert Guillem Balague believes he would be the right choice for Arsenal.
"Luis Enrique seems to be perfect for many reasons,” Balague said. “One: because he doesn’t care what you think or what I think or what nobody thinks. He’s got a very clear idea of what he has to do, not so much with the style but the winning as well, but in a certain way.
“He would give mental strength and decisiveness in terms of decisions. He wouldn’t care about a star who earns more, he would just go for the kill and do what needs to be done."
Whoever eventually is chosen as the successor for Wenger will reportedly only have a £50m budget to work with in the summer, according to the Mail's report. As the Gunners look to take on the challenge of competing with the likes of Manchester United and Manchester City next season.