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.”