The Fiorentina wide-man starred for his side in Serie A last term, helping them to secure Europa League qualification.
Since then La Viola have bought out the 50% of his rights owned by Udinese – a common thing in Italian football – and are ready to cash in.
The Florence outfit are said to be demanding a big sum for their man, to which United have replied with a £32m offer.
Louis van Gaal is known to be chasing attacking talent to boost his squad this summer after taking over from David Moyes following the club’s disastrous 2013/14 campaign.
Arsenal were thought to be the English side most likely to snap up the 26=year-old, but now the ball seems to be in United’s court, even though Fiorentina want a little more cash for their prized asset.
The pacey Cuadrado has thrived since being converted to a wing position after breaking through as a full-back, and has been starring for his nation at the World Cup on the left flank.