Chelsea wants to break the British transfer fee record for a striker it has already sold once before
The Sun's report adds that Chelsea may even throw its own striker, Loic Remy, into the mix to tempt Everton into making the deal, and it wants a firm answer by the weekend.
But Everton is playing hardball, demanding £75 million for Lukaku - almost three times the £28 million it paid Chelsea for him in 2014.
Lukaku would be an interesting addition to the Chelsea manager Antonio Conte's team. As a left-footer he is versatile on both sides of the wings and is known for his physicality in getting past defenders.
The current English football club transfer fee record is Manchester United's purchase of Ángel Di María from Real Madrid in 2014 for £60 million, so even Chelsea's initial offer would exceed that.
But both could soon be beaten by United's likely signing of Juventus' Paul Pogba, which is rumoured to be £100 million - a world record that beats Real Madrid' signing of Gareth Bale for £85 million.
Coincidentally, United would also be buying back a player who played for them already - Pogba spent a year playing for the club in 2012 before Alex Ferguson decided not to renew his contract.
Lukaku is not the only massive sale Everton could make before the new Premier League season starts - rumours continue that Manchester City is also interested in buying John Stones, though not for the hefty £50 million price tag specified in Stones' release clause.
Chelsea's first Premiership game is at home against West Ham August 15, while Everton plays Tottenham on August 13.
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