The Belgian striker has allegedly informed Koeman that he sees his foreseeable future at Goodison Park.
Lukaku has been linked with a host of top clubs from around Europe this summer after the best goalscoring season of his career, which he scored 25 times in 46 appearances in all competitions.
The 23-year-old has emerged as one of the most sought-after strikers in European football, and had been linked heavily with a return to former club Chelsea.
However, the Mirror claim that he has instead informed Koeman that he wishes to remain at Everton this season, in what is an incalculable boost for the Merseyside clubs.
Things have changed a lot at Goodison Park since the latter weeks of last season, with Roberto Martinez replaced by Koeman and Farhad Moshiri's takeover injecting funds into the club.
Yannick Bolasie was signed from Crystal Palace for £25 million, and the Toffees are prepared to make Lukaku the highest-paid player in their history.
It now appears that Lukaku is set to commit his future by signing a lucrative long-term, worth over £100,000 per week according to the Mirror's report.
That will not only end speculation about his future, but only serve to increase his market value.
Everton are likely to continue their hunt for another centre-forward, however, with Koeman uninterested in reintegrating £13.5 million January signing Oumar Niasse.