Lukaku, 28, is at the head of Chelsea’s search for a new forward after Borussia Dortmund ruled out selling striker Erling Braut Haaland this summer amid interest from the Blues.
While Inter’s owners are wrestling with major financial issues, they are determined to keep hold of the Belgian.
Lukaku, a former Chelsea player, was a pivotal figure last season as Inter won their first Serie A title since 2010.
It is understood Chelsea were willing to include Spanish defender Marcos Alonso in the deal.
It remains to be seen whether the Champions League winners will make a higher offer, although Inter believe they are capable of resisting it and Lukaku has previously spoken about his intention to stay at the club.
Lukaku played 15 times during three seasons at Chelsea, between 2011 and 2014, but was sold to Everton after an impressive season on loan.
He came close to rejoining the club in 2017 but instead opted to link up with Jose Mourinho at Manchester United for £75m.
Lukaku scored 42 goals in 96 games for United, but left for Inter in 2019 for a club record fee of 80m euros (£74m) on a five-year deal.
Lukaku has scored 47 goals in 65 Serie A games for Inter Milan