I didn't read about the salary that was offered to him, but it was probably substantially higher than he would have gotten either at Chelsea or at Inter Milan. But I think he is in a situation in his career where money still isn't everything and he wants to play a few more seasons in the Champions League and win some titles if possible. Not some titles in the Champions League, but in general. I guess he thinks that there were some misunderstandings in recent years that he wants to clean out from his chronology as a player. Inter Milan would be the best destination for him to do that and I still think that both the club and the player will do what they can to sort that out.
If we check statistics in Serie A actually Lukaku did not contribute much to Inter from 25 matches Lukaku scored 10 goals and 6 assists but in the Champions League a little better 8 matches 3 goals overall not too impressive. Lukaku's best career in 2020/2021 when he got several awards after moving to Chelsea his career seemed dim and now he is 30 years old so I thought he only had 3 years to maintain his career, besides the Saudi League and Inter who else wants to approach Lukaku? I have never heard of another club want to bring Lukaku.
You are wrong and I think Lukaku had a good performance in the last games of the previous season we saw, however, because of his age they can't count on this player as a long-term plan but I think Lukaku could even have a better performance if he didn't get injured because Lukaku helped Inter Milan a lot in the champions league and he likes to play in this team that's why I think investing on Lukaku is not a bad move.