Thomas Tuchel's side offered around £88million plus Marcos Alonso for the Belgian, but the offer was turned down by the Serie A champions, who have insisted they need Lukaku to remain at the club.
Chelsea are keen to bring in a new striker and have turned their attention to other targets after Borussia Dortmund insisted they wouldn't part ways with Erling Haaland.
Lukaku played a key role in Inter's title win last season, scoring 24 goals, and has insisted he is happy at the club, and is fully focused on the new campaign.
"I did get to relax, but I've also been training to stay fit and be able to start [the season] well," he told the club's official media channel last week when asked about his summer break.
"We now have a few weeks to prepare as well as we can for the new campaign. We hope to do well and achieve even more than last year. I'm thrilled to be back. We hope to continue down the path that we've started and to keep winning. All the players will work hard to make your dreams come true."
Lukaku previously spent three years at Chelsea, making just 15 appearances, failing to score a goal, and moved to Everton for £28m in July 2014. He then moved to Manchester United for £75m in 2017, before eventually joining Inter for £73m two years ago.