The north London club have begun searching for a new manager after Arsene Wenger announced his intention to step down at the end of the season after nearly 22 years in charge.
A host of names – from Brendan Rodgers to Mikel Arteta – have been linked with the hotseat at the Emirates and former Barcelona boss Enrique is the latest.
Chelsea are believed to be keen on appointing Enrique and will offer him as much as £10m-a-year to move to Stamford Bridge.
However, Arsenal believe they are the more attractive option for Enrique and the Gunners believe they would beat Chelsea to the Spaniard if they opt to make him an offer.
Arsenal want their next manager to be in the mould of Wenger but to have a greater appreciation of modern football.
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Massimiliano Allegri has been considered but theres fears over his playing style, while Carlo Ancelotti is expected to become Italy boss.
Ivan Gazidis confirmed on Friday that Arsenal will announce Wengers successor as soon as possible but the Frenchman will be keen to focus on the clubs Europa League campaign.
The Gunners take on Atletico Madrid in the first leg of their Europa League semi-final on Thursday.