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Konate is only tied down at Liverpool until the end of the upcoming season and is yet to sign a new deal.
A move to the Bernabeu appears to be a real possibility, given that Real Madrid usually get their own way in the transfer market.
They were initially expected to wait until next summer, when Konate will become a free agent if he decides against extending his stay at Liverpool.
However, it seems the La Liga heavyweights are now looking to sign him this summer.
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They reached out to the Reds earlier this week to discuss their interest in bringing Konate to the Spanish capital, according to L'Equipe.
The report states that Real Madrid asked about the kind of money that would be required to secure a deal.
Liverpool chiefs are said to be open to the idea of selling Konate, rather than losing him for nothing in a year's time.
Real Madrid could simply wait until Konate's contract expires before snapping him up on a free transfer, meaning it could be in Liverpool's interest to take whatever they can get.
Real Madrid used a similar tactic to land Trent Alexander-Arnold earlier this summer.
They ended up paying just £10m to secure his services in time for the Club World Cup, with Liverpool having little choice but to accept a low-ball offer.
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