The Telegraph says Arsenal chiefs are split on who should succeed Wenger – with Enrique a frontrunner for the job.
Chief executive Ivan Gazidis, head of football relations Raul Sanllehi and head of recruitment Sven Mislintat are all leaning towards different candidates, with Wenger expected to leave this summer after 22 years at the club.
Former Dortmund scout Mislintat is said to be keen on raiding Germany to find the Gunners' next manager.
Mislintat was pictured in the stands for Schalke's 2-0 win over Dortmund on Sunday, with victorious boss Domenico Tedesco under the microscope.
Hoffenheim manager Julian Nagelsmann, 30 and two years younger than Tedesco, is also in the Mislintat's sights.