Only appointed in June, Lopetegui is fighting for his job after a poor run of form has seen them fail to win any of their last four La Liga games, losing three of them.
But while ex-Chelsea boss Antonio Conte appears to be the overwhelming favourite to take the reins, a report from The Sun claims that Real still prefer Pochettino after failing to attract him during the summer as a replacement for Zinedine Zidane.
It is said that Real are 'double checking' to see if there is any chance they might be able to get Pochettino before making a final decision on hiring Conte.
The likelihood of the Argentine leaving Spurs mid-season seems slim at best. He only recently signed a new five-year contract and the cost of releasing him from that deal would be vast for Real.
Rather bizarrely, the Real manager saga is believed to have taken another twist after Conte's brother denied there has even been any contact over the possibility of taking over at the Bernabeu.
Despite recent speculation suggesting that Conte had heard from Real and let them know of his immediate availability, sibling Daniele Conte, who also acts as his agent, says it is not true.
"Antonio is currently on vacation and has not received any phone call from Real Madrid. We can certainly say that it is a great club, but for now there have been no contacts," Daniele declared while speaking on Spanish network Cadena Ser this week.