The Portuguese coach has emerged as a leading candidate within the club, with sources stating that president Florentino Perez is understood to be well aware of his desire to return to the Bernabeu.
Mourinho, who previously managed Madrid between 2010 and 2013, is believed to be open to taking charge again this summer.
He was already a preferred candidate of Perez but that willingness has put him in an even stronger early position, as the club want to move quickly with their plans for next season.
Madrid’s hierarchy are keen to avoid delays, with key decisions looming over the squad.
Reinforcements in central midfield and central defence are expected, while there are also internal matters to resolve regarding returning players and the overall balance of the squad.
Klopp, meanwhile, represents a high-profile alternative that could yet emerge. So far, there is more uncertainty attached, as Madrid do not yet have a clear indication that he would be willing to step away from his current role within the Red Bull football structure and return to day-to-day management.
Should Klopp signal that intention, he would immediately become a serious contender and could turn the situation into a genuine head-to-head battle.
There are potential complications surrounding a Mourinho return, particularly following recent tensions involving Vinicius Junior. However, sources suggest those issues are not viewed as a major obstacle at this stage.
Madrid ideally want clarity on their next manager before the end of the season, underlining the urgency behind the process.
As it stands, the picture is becoming clearer that they have two elite candidates in mind, but with Mourinho just positioned as favourite - for now.