Brazilian Leonardo stepped down as sporting director on Tuesday, along with head coach Gennaro Gattuso, amid reports of financial cutbacks at the seven-time European champions.
The former European heavyweights finished one point behind Inter Milan in Serie A in the battle for the fourth and final qualifying spot.
Gazidis said in an interview with Gazzetta Dello Sport, that with the help of Maldini, they could "bring Milan back to the top".
Maldini retired in 2009 after a glittering 24-year career during which he won seven Serie A titles and five Champions League crowns. The defender was also Milan's record appearance holder with 902 matches.
The 50-year-old returned to the club last August as their sporting strategy and development director.
"I want Maldini as the technical director. I admire him very much," former Arsenal chief executive Gazidis told Gazzetta.
"He represents the values and the culture of the club.
"I would really like Paolo to stay with us and help me in this great challenge with an increasingly central and important role. He is ideal for managing the technical area.
"Paolo asked for time to reflect and understand if he has the right energy to start over with this difficult and arduous project, which requires his utmost commitment and which he must totally buy into. He must feel 100 percent."
Maldini would also have a role in the choice of a replacement for Gattuso, another emblematic former player who took over in November 2017 and was under contract for another two years.
"The new coach will be chosen with care," said Gazidis.
"Their age or nationality doesn't matter. What matters is that they're suited to what we want to do and build.
"The choice of coach, I hope with Maldini's help, will be made calmly, studying every detail: history, character, profile, results, statistics. We don't want to do anything wrong."
AC Milan are adopting spending curbs faced with mounting debts and threatened with UEFA sanctions for breaking financial regulations.
Gazidis, 54, said that Elliott Management had saved the club from bankruptcy, as its Chinese owner failed to replay debt owned to the US-based hedge fund.
"Elliott saved the club by injecting them with EUR220m of capital, otherwise they would have gone bankrupt with the risk of relegation," added the South African-born businessman who arrived from Arsenal last December.
AC Milan have not qualified for the Champions League since the 2013-2014 season.