The Wales international has been heavily linked with a return to the Premier League in recent months.
Manchester United and Chelsea have been linked with a move for the 28-year-old winger.
And according to Spanish outlet Diario Gol, Real president Florentino Perez is keen to offload Bale this summer.
They say the chief is putting pressure on Zinedine Zidane to sell the former Tottenham star to free up a move for Monaco sensation Kylian Mbappe.
Diario Gol add that Chelsea and United are desperate to land Bale but Real's asking price remains a problem.
Real want at least the £85m they paid for Bale back in 2013.
However, Diario Gol say Chelsea and United have told Real he isn't worth that with the winger's injury woes over the last two seasons.
Mourinho has been vocal about signing Bale this summer, with the United boss claiming he'll make a move if Real want to sell.
He said: "If he plays [against United in the Uefa Super Cup tomorrow night], it is the clearest signal that he is staying there.
"Then I won't even think about it. If he is not in the club's plans and it is true that a player like Bale is at the departure gate, I will try to be there waiting for him at the other side."
