Speaking to Britain's Daily Star newspaper, Lukaku did not rule out remaining with the Toffees but said "I want to win titles" in a hint that he sees his future away from Goodison Park.
Having joined Chelsea from Anderlecht in 2011, he never managed to win a regular starting spot and moved to Everton, initially on loan, in 2013.
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"I'm at a point where I have to make a good career choice," Lukaku is quoted as saying in the Daily Star.
"We have a new investor at Everton and out of courtesy I will listen to what he has to say. But I have my own ideas in mind. I want to win titles."
Lukaku scored 18 goals in the Premier League in 2015-16, his best return yet as he finished fourth in the scorers chart, despite Everton struggling to a disappointing 11th-placed finish.
"I have had a very good season, but it is time for me to write myself a CV. That is why I play football," added the 23-year-old.
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"I got that trophy-winning mindset from Chelsea.
"People in Belgium don't see me playing for Everton, but I am respected by fans, managers and experts like Jamie Carragher, Graeme Souness and Thierry Henry."
Lukaku has been linked in British media with a return to Chelsea, who will be led by current Italy coach Antonio Conte next season.
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