The Mirror says United are aware of Valverde being unsettled in Madrid after failing to be convinced by coach Xabi Alonso's methods.
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The Uruguay international is no longer a first-choice and has been pushed to accept being played as a wing-back by Xabi.
Valverde situation is being watched by United, with intermediaries alerting them to the midfielder's potential availability.
Real Madrid would consider selling Valverde for £100m, though United rate the player closer to £70m.