Fabinho took his time to get going at Anfield last season following a big-money move from French side Monaco.
But he ultimately established himself as a midfield lynchpin and was key as they won the Champions League in June.
Real have been tipped to prise anchorman Fabinho away from Liverpool as they continue their summer spending spree.
Colombian journalist Jose Luis Vergara claimed on Twitter on Tuesday that a switch is set to be announced.
“TOTAL Agreement between Real Madrid and Liverpool,” Vergara said. “FABINHO will be player of #RealMadrid before market close.
“He will sign until 2024. More information in the next few minutes. The Brazilian player returns to what was his home for a season.
“The transfer will be made official when the #Liverpool [sign] a replacement. The Spanish club will pay 65M € + 10 in variables, and the player’s salary will be 7.5M € annual.”
However, The Athletic’s James Pearce has rubbished the speculation with a simple two-word statement.
Asked to stop the transfer talk in its tracks, Pearce simply tweeted: “Absolute nonsense”.
Fabinho started on Sunday as Liverpool took on Premier League champions Manchester City in the Community Shield.
But he was unable to help Jurgen Klopp’s side secure more silverware as City ran out winners on penalties following a 1-1 draw.
The 25-year-old is in line to figure on Friday when Liverpool kick off the new Premier League season by facing Norwich.
Klopp insists Liverpool must not worry too much about their hectic schedule as the campaign gets underway.
“I’m fine with Friday and I’m fine with the time we have until Chelsea [in the UEFA Super Cup],” Klopp said.
“Am I overly happy that we then have only three days for the Southampton game? No. But it’s the situation and I cannot change it, I don’t even think about it.
“We come back from Istanbul and then play on the Saturday. We then play Southampton. All these games will be incredibly difficult but it’s our schedule and we never moaned about the schedule.”