Lukaku, who scored 24 Serie A goals last season to help Inter win their first league title since 2010, was sold to Chelsea last week for a reported £97.5million (€115m).
Inter needed to sell players as they deal with financial difficulties compounded by the coronavirus pandemic, and they had already sold full-back Achraf Hakimi to Paris Saint-Germain.
Meanwhile, Lukaku's former strike partner Lautaro Martinez has been linked with a move to London as well, with Tottenham reportedly agreeing a fee for the Argentina forward, though that move seems likely to depend on Harry Kane's future.
In the wake of Lukaku's departure, Inter have bolstered their squad with the arrival of experienced striker Edin Dzeko, from Roma, and Netherlands right-back Denzel Dumfries, who has arrived from PSV. The Nerazzurri had previously signed Hakan Calhanoglu on a free transfer from rivals Milan.
Inter start their title defence against Genoa on Saturday and, ahead of his first league game in charge, former Lazio coach Inzaghi revealed the club did all they could to keep Lukaku, who is poised to make his second Chelsea debut on Sunday against Arsenal.
"It was an unexpected and unplanned departure," Inzaghi told a news conference on Friday, also confirming Inter are in the market for another forward.
"It is normal that I, [chief executive Giuseppe] Marotta and [sporting director Piero] Ausilio tried to convince him.
"In the week I trained him, I understood why he is so decisive and loved. When he decided, he came to tell me honestly that Chelsea was his dream. We talked to him but the choice was made by the boy. With Dzeko and another signing, we will cover his departure."
Asked about Inter's ambitions for 2021-22, Inzaghi said: "I think that Inter's goals are certainly ambitious.
"I don't like making proclamations, but what will make the difference will be the motivations of the boys: during the retreat they showed me that they have a lot of them, I'm confident.
"We will do everything to ensure that things go the right way. It will obviously be a season with difficulties, but the goal in all of us is the match against Genoa.
"There is an excellent relationship with the club, a team level is still missing something but so far I am happy with the players who have arrived.
"We know that there have been some sales, such as that of Hakimi, that of Lukaku and the problem of [Christian] Eriksen. Dzeko would have been my request regardless of Lukaku's situation. Dumfries will need to settle down."