German Heynckes, who turned 68 this month and speaks fluent Spanish, is leaving Bayern at the end of the season after leading the club to the Champions League and Bundesliga titles.
He guided Real to a Champions League crown in 1998 and has also coached La Liga clubs Athletic Bilbao and Tenerife.
Heynckes' agent, Enrique Reyes, told Spanish radio it was up to Real president Florentino Perez to decide whether to hire his client or continue his pursuit of Paris St Germain's Italian coach Carlo Ancelotti.
"It is Florentino's decision," Reyes said. "If Florentino calls us we would accept the offer.
"The one who has to accept is not Heynckes it is Florentino. For the moment, he (Heynckes ) is only retiring from the Bundesliga."
Mourinho is leaving Real after the club ended the season, his third in charge, without major silverware, while Heynckes is being replaced at Bayern by former Barcelona coach Pep Guardiola.