Leonardo left the role, departing alongside head coach Gennaro Gattuso as the Rossoneri look to rebuild after again falling short of Champions League qualification.
Lille director Luis Campos has been linked with Milan, but chief executive Gazidis is instead hoping to appoint club great Maldini, the team's sporting strategy and development director.
However, Maldini - Milan's record appearance holder with 902 matches across 24 years at San Siro - has been allowed time to make a decision.
"I admire [Maldini] immensely," Gazidis told Gazzetta dello Sport. "He represents the club's values and culture. You see Maldini and you see Milan.
"He is ideal for a technical role. He would have a good staff around him. It wouldn't be a role just for show but one that is absolutely central in technical decisions.
"I'm not asking him to do everything by himself. He'll work as part of a group, he'll be supported and helped. He can give us a lot.
"It's been a tiring season for everyone and Paolo has asked for time to reflect and consider if he has the energy to start over with this difficult and arduous project.
"It will require his utmost commitment - he must totally buy into it. He must feel 100 per cent. That's the condition for anyone who wants to come to Milan. But in the case of Maldini, he is Milan.
"I don't want to think about people who are not Maldini. I'm waiting for Paolo's decision, then we'll make a decision on the coach and the players."
With Gattuso also having departed, Gazidis plans to be able to call on Maldini to help with the next appointment.
"The coach will be chosen with care," he said. "Their age or nationality doesn't matter. What matters is that they're suited to what we want to do and build.
"The decision - hopefully with Maldini's help - will be made calmly, studying every detail: history, character, profile, results, statistics. We don't want to do anything wrong."