The Daily Mail claims Real officials have made initial contact with representatives of Pochettino, who has refused to rule out a return to Spanish football.
Madrid coach Zinedine Zidane is under intense pressure after the club were knocked out of the Copa del Rey by Leganes on 24 January at the Bernabeu.
The Spanish giants won the La Liga title and a record 12th European Cup under the Frenchman last season, but have struggled to replicate that form this time around and are trailing arch rivals Barcelona by 19 points in the league.
Zidane described the Leganes defeat as his "worst moment as a coach" and admitted that his future at the Bernabeu depended on how the club fared in the Champions League – Real host Paris Saint-Germain in the first leg of the round of 16 next month.
"I'm responsible for this, I'm the boss. I have to find the solutions," the Frenchman was quoted as saying by the BBC. "We've found some of them but not consistently enough. I need to take responsibility and I'll always fight."
Pochettino is reported to be Madrid's first choice to become head coach should they decide to dispense with Zidane at the end of the season.
The former Espanyol coach said this week that he is fully committed to Spurs for the moment, but refused to rule out managing Real in the future.
"My commitment is massive [to Spurs]. I am working like I am going to be here forever," Pochettino was quoted as saying.
"But in the end you never know what is going to happen in football.
"Tomorrow [Tottenham] chairman Daniel Levy could have a bad night and say, 'Oh I am going to sack Mauricio', then I look stupid saying I am not going to work in one place or another or another."