Ronald Koeman is planning to add a striker to his squad to partner the Belgian international.
The striker had been expected to depart Goodison Park after openly discussing his desire to play for a Champions League club.
It was thought there was interest in him too, with his 25 goals in all competitions last season catching the eye of teams around Europe.
However, Everton played hardball over their star man and slapped a £65 million asking price on the 23-year-old to ward off any potential suitors.
As a result, the Blues are yet to receive any offers for the Belgian international and the Liverpool Echo reports there is a growing optimism around the club that he will stay at Goodison Park.
New Everton manager Ronald Koeman is also keen to do as much as possible to convince him to do just that, and is reportedly looking for a striker to partner Lukaku next season
Lukaku often cut a frustrated figure for the Toffees last season as he was left isolated alone up front and the Dutchman is keen to rectify that.
He also needs sufficient back up for the striker, with Arouna Kone and Oumar Niasse failing to prove they can be an effective alternative to the Belgian last season.