The Welsh winger has been heavily linked with a return to England at the end of the season but is determined to remain in Spain, according to the Daily Mail.
It is thought that both United and Chelsea were hugely optimistic that Bale could be tempted to come back to the Premier League after coming in for criticism in Madrid in recent weeks.
Bale's car was attacked by fans disappointed by his performance in the Clasico against Barcelona on Sunday night.
But the former Tottenham star has now reportedly told United, City and Chelsea that he is keen to remain in La Liga and win over his detractors.
GETTY
Gareth Bale wants to stay at Real Madrid
I don't get the impression that Gareth is someone who is fazed by these sorts of things.
Joe Allen
Bale will play for Wales against Israel in a Euro 2016 qualifier on Saturday and Liverpool's Joe Allen says he has not been affected by a difficult week.
He said: "He's the same as usual. He's a world-class player who shows that in every minute in every session.
"It is great to have him here. I doubt the Israelis will be looking forward to facing him at the weekend.
"I don't get the impression that Gareth is someone who is fazed by these sorts of things. It comes with the territory at a club like that. I doubt it has too much influence on him."
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