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The Belgium striker only scored eight goals in 26 Premier League appearances for the London club in the 2021-2022 campaign.
Lukaku found the back of the net 64 times in 95 games during his first spell with Inter, winning the Serie A title last year before signing for Chelsea for a second time.
The 29 year-old, who has picked 90 as his new shirt number, had made no secret of his desire to play for Inter again and feels he is back where he belongs.
"It's like coming home. My family and I were so happy here thanks to the people, the fans and my team-mates," he said. "From the first day I arrived here, everyone helped me settle in.
"I'm very happy. I kept my home here when I went to England, so that means something. I'm pleased to be back and now I can't wait to get out there with my team-mates."
Belgium's record goalscorer is determined to put a miserable season with Chelsea behind him.
He added: "It's an amazing feeling. What we achieved here a year ago was fantastic. That goes for the team, the fans and also me personally.
"Inter have given me so much and I hope to do even better than last time. It's now time to work hard as a squad and hopefully do even better than before."
Lukaku also expressed his gratitude to Inter president Steven Zhang for getting the deal done.
"We spoke about so many things and how it seemed impossible for me to come back here, but we managed it and I'm grateful to him for the way he kept pushing and believing it was possible," he said. "We managed to do it in the end and I'm delighted to be back."