Several British newspapers reported on Tuesday that the Reds had made a £62 million offer for Roma and Brazil No1 Alisson, but that the Serie A outfit wanted £66.5m.
Either figure would be a world record transfer fee for a keeper.
And Real, who have long sought a new stopper to replace Keylor Navas and were a broken fax machine away from signing Manchester United’s David De Gea in 2015, are reportedly close to wrapping up a £31m deal for contract rebel Courtois.
The Belgium international, who won the World Cup’s Golden Glove award for his inspired displays in Russia, has not signed an extension at Chelsea with his contract due to expire in 2019.
Courtois’ children live in Madrid, where he used to play for Atletico.
“The only thing I’ve always said is that after the World Cup I look at everything,” he said earlier this month. “The only thing is that my children are the most important and I’ll make the decision that best suits me for everything.”
And according to French radio station RMC, the Blues are willing to accept Los Blancos’ offer and cash in on the 26-year-old before he is free to leave next summer.
The London club will now almost certainly be in the hunt for a new keeper themselves and the development could spark a transfer battle with Liverpool for the right to sign Alisson.
The 25-year-old has been with Roma since joining from Brazilian side Internacional in 2016 and made 37 Serie A appearances last season.
He played in all five of Brazil’s games at this summer’s World Cup, keeping three clean sheets.
Liverpool keeper Loris Karius was back in the spotlight last week after a 3-2 friendly win against Tranmere at Prenton Park saw him spill a free-kick which allowed the hosts to score.
Karius was at fault for two of Real Madrid’s three goals when they beat Liverpool 3-1 in May’s Champions League final, after which it was revealed the 25-year-old German was suffering from concussion.
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