Liverpool are said to be considering making a move for the youngster if they sell Philippe Coutinho.
Diario Gol claim Ceballos held a meeting with the Madrid board prior to the Clasico last week as he looks to determine his future.
Liverpool are believed to be weighing up their options with Coutinho continually linked with Barcelona and Sergi Roberto not available in part-exchange.
And that has seen Klopp target Ceballos with a massive hole potentially being opened up by Coutinho’s prospective departure.
GETTY
Spain Under-21 international Dani Ceballos joined Real Madrid from Real Betis this summer
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Whether Real would be willing to let the 21-year-old move on during the transfer window remains to be seen.
Diario Gol claim the most likely scenario would be a loan switch to Anfield despite Liverpool potentially selling Coutinho for around £150m.
Liverpool have already agreed to sign Virgil van Dijk in January, with a world record fee for a defender signed off.