The Spanish and European champions were reported to be readying a mega-money bid to sign teenage Monaco striker Kylian Mbappe and there were suggestions that Bale would be sold to finance the move.
But Real officials have assured Bale that he is the future of the club and the long-term successor to Cristiano Ronaldo, according to the Observer.
Real boss Zinedine Zidane remains interested in signing Mbappe, but a move is not expected to happen until next summer at the earliest.
The Spanish giants had a £103m bid turned down for the 18-year-old striker earlier this summer, with Monaco holding out for a fee of £178m.
United manager Jose Mourinho said before United's pre-season friendly against Real that he was interested in Bale but that there was little point to pursuing him.
"No [he was not on my list of targets]. No, because you have contacts, you have feelings, you understand things," the Portuguese was quoted as saying by the Independent.
"I think it was clear that Bale likes Madrid, likes the challenge, likes the situation. He's in a club in a very good situation now."
Bale has said he is focused on featuring more regularly for Real in the upcoming season after playing in just 19 La Liga games last campaign.
The former Tottenham player missed several crucial fixtures in the final six weeks of Real's campaign with a calf injury, but returned to make a cameo in the club's Champions League final victory over Juventus in Cardiff.
He spent three months on the sidelines earlier on in the season due to an ankle injury.
"My family is happy and I am happy so, yeah, we will continue what we are doing. We are winning trophies and I am happy," Bale was reported as saying by FourFourTwo in June.