According to David Woods of the Daily Star, Gazidis was "blown away" with Allegri after he engineered Juventus' comeback against Tottenham Hotspur in the UEFA Champions League. His side fought from 1-0 down in the second leg to win 2-1 and progress with an aggregate score of 4-3.
Gazidis was also impressed with "the strong, collective spirit" Allegri's side displayed at Wembley Stadium, and the Italian has been learning English over the last 12 months.
The 50-year-old changed the shape of his side midway through the second half to a back four, and Bleacher Report's Sam Tighe pointed to that moment as key:
Within minutes of Allegri's tactical change, Juve scored twice in quick succession through Gonzalo Higuain and Paulo Dybala before holding on for the win.
Juventus analyst Arjun Pradeep noted the significance of the victory:
Allegri's flexibility and willingness to adapt his tactics are qualities Wenger lacks, with the Frenchman often rigidly persisting with his approach to a game even when it's not working.
The winning mentality the 50-year-old has imbued in his side would also go a long way at Arsenal, who—despite winning three FA Cups since 2014—frequently seem to lack in high-pressure games.
Allegri would also bring with him a highly impressive CV to the Emirates Stadium.
He won the Serie A title with AC Milan in 2011—their only scudetto since 2004—and he has completed the league and cup double in each of the last three seasons with Juventus while reaching the Champions League final in 2015 and 2017.
Juve have faced a stern challenge from Napoli in Serie A this season, but after beating Udinese 2-0 on Sunday and Partenopei drawing with Inter Milan, the Bianconeri are now a point clear with a game in hand.
Football journalist David Amoyal noted their impressive form:
Landing Allegri would be an incredible coup, though it may be difficult for Arsenal if they don't manage to land Champions League football. Their only realistic path to doing so is to win the UEFA Europa League too, narrowing their margin for error.
It will be reassuring for Arsenal fans that the club is aiming high as they consider replacements for Wenger, but securing his signature is another matter entirely.