Inter director Beppe Marotta said the club contacted Massimiliano Allegri in the summer before contracting Simone Inzaghi as coach.
The Nerazzurri were looking for a replacement for Antonio Conte, who left his post shortly after the end of the season in which the club won their first league title in 11 seasons.
Allegri, who was a free agent, was heavily linked with the role as rumours began to grow that Conte could leave, while Inzaghi was still Lazio coach.
“Before getting Inzaghi, we had contact with Allegri,” Marotta told DAZN. “We never thought Simone would be available, while Max and obviously had an interesting profile so we spoke.
“We called him [Inzaghi] not knowing he was at dinner [with Lazio president Claudio Lotito] and Simone was obviously a little embarrassed.
“I have to say that the timing and intuition by Piero Ausilio and myself were decisive in getting him to find a quick agreement but with respect towards Lotito.
“When a coach or a player stay at a club for too long, as was the case Inzaghi at Lazio, it is right to try something different, an opportunity to grow.”
Inzaghi had been at Lazio as coach since 2016 and enjoyed a successful period as he led the club to two Supercoppa successes as well as 2019 Coppa Italia.
Allegri, who had already managed Milan from 2010 to 2014, had been available since leaving Juve in 2019 before returning to take charge of the club for a second spell at the beginning of the season.
Marotta worked with Allegri at Juve, having brought him to the Turin in 2014.