With just over a year to run on his deal at Anfield, Konate could run down his contract at Liverpool and leave next summer in a Bosman arrangement for Real Madrid.
Marca says Real chiefs are watching developments, which could see them prepare a pre-contract offer for Konate to consider in less than six months' time in January.
France international Konate, meanwhile, said at the end of last season: "You know, I just want to say one thing: Everything you saw on social media is completely fake.
"I didn't even ask for something from the club. They made me an offer, and I replied to them with something, that is it. The rest (speculation) that I want to wait until the end...I want this, all of it is completely fake.
"And even the big journalists said in November I am very close to signing the deal. I was like: 'Ah, they have more information than me!?' Because it was not true at that moment and that is why I was a little bit mad because it was not true at this moment.
"I knew at one moment people would think I didn't want to sign the contract but I was never close to signing it. We will see what will happen.
"Progressing? Not really but my agent met the club last week but I don't know what will happen. I cannot say what my agent said to me but yeah, we will see what will happen."