According to unnamed Spanish sources cited by talkSPORT, the La Liga title hopefuls are plotting a switch of their own for the German international playmaker and could be willing to offer him the opportunity to move to the club this summer or, alternatively, when his current contract at the Allianz Arena runs out in 2015.
24-year-old Kroos - who joined Bayern in 2006 - has been in good form for the Bavarians this season, yet has found his future the subject of fervent press speculation with Manchester United heavily linked with a summer move for a player who appears to be valued at around the £40 million mark.
Barcelona have also been mentioned with regard to a potential transfer and such gossip appears to be driven by the player still evidently embroiled in somewhat of a saga with his current employers regarding the possibility of a contract extension.
With Kroos still not having put pen-to-paper on a new deal to remain with the reigning European and Bundesliga champions, Bayern may well be forced to sell him for the best possible fee during the next transfer window in order to avoid the possibility of losing him for nothing a year later.
Whether Manchester United can maintain any realistic hope of signing the player with Real Madrid now seemingly circling remains to be seen.
While there is little to choose between the two clubs in terms of global profile, stature and reputation, Manchester United's likely absence from the Champions League next season - unless they can defy the odds to win this year's competition, of course - could surely have some bearing on which club Kroos decides to pledge his future to.
Real Madrid - who currently sit third in La Liga and are fighting for supremacy with their two biggest rivals in Barcelona and Atletico amid one of the most hotly-contested Spanish title races in years - have made it a policy to complete at least one major signing in each summer since club president Florentino Perez arrived at the Santiago Bernabeu and Kroos would certainly fit that bill.