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The centre-back's current deal is due to expire in 2026 and he is yet to agree terms over an extension.
The report claims that Konate has 'already started the countdown on his contract' and has rejected Liverpool's latest offer of a new deal, with club chiefs fearing a repeat of the Alexander-Arnold saga.
Konate spoke out about his future after Liverpool's final Premier League game last month, admitting he is unsure what will happen.
"You know, I just want to say one thing: Everything you saw on social media is completely fake," he said.
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"I didn't even ask for something from the club.
First thing, I am very happy because we have won the Premier League trophy and we lifted it today.
Madrid are eyeing a move for Ibrahima Konate(Image: Andrew Powell/Liverpool FC via Getty Images)
"We have to enjoy it, have a rest and hopefully next season will be better than this.
To be fair, to win a trophy, even when I won my first trophy with Liverpool, the FA Cup, was just amazing.
"And today, to win the Premier League, I know I have a lot of objectives with this club: to become a leader, to win everything.
But I know where you want to go with this question and this is another conversation with the club and my agent, we will see what will happen in the next few weeks or I don't know when but we will see, let me say that.
"Progressing?
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