The duo are understood to be Arsenal's top two choices to take over as boss this summer when Arsene Wenger steps down from his role after 22 years at the helm.
Luis Enrique has been out of work since leaving Barcelona following a spell which saw him win nine trophies in three years, while Allegri is on the verge of leading Juventus to an unprecedented seventh consecutive Serie A title.
However, Sky Sports News reports that both candidates are unsure about working so closely with Arsenal's head of football relations Raul Sanllehi and head of recruitment Sven Mislintat, both of whom were appointed last year.
The likes of Carlo Ancelotti, Mikel Arteta and Liverpool assistant manager Zeljko Buvac have also been linked with the soon-to-be-vacant post at the Emirates, with Arsenal looking for a new manager for the first time since 1996.
Wenger took charge of his final home game on Sunday, with Arsenal running out 5-0 winners against Burnley.