Bale has impressed in La Liga but struggled for regular match action towards the end of last season.
Zinedine Zidane has preferred to utilise Isco in an attacking role with Bale increasingly left on the bench.
If Real sign Kylian Mbappe, Bale could find his first-team options even more limited.
Spanish news outlet Diario Gol say Paris St-Germain have made enquires regarding Bale's availability.
It's suggested Real are keen on him moving to Paris because it would effectively rule the Ligue 1 giants out of a move for Mbappe.
But Bale's agent, Jonathan Barnett, has told Real's top brass that his client won't move to France.
Diario Gol say he's instead given the green light to an Old Trafford switch.
Real paid around £85m to secure his signature in 2013.
It's suggested he could available for around £70m this summer.
Real, however, realise his marketing appeal for United and could demand much more.
United have only signed Victor Lindelof from Benfica this summer.
They're also closing in on a £75m deal for Everton striker Romelu Lukaku.
