The paper adds that Paris Saint-Germain have already matched that offer as they look to land the Argentine star - however reports in France say the are ready to make an improved offer of €70 million in order to get their man.
Real Madrid are reportedly looking to sell Di Maria, who will miss Argentina's World Cup semi-final clash against Holland on Wednesday through injury - in order to finance a move for Monaco sensation James Rodriguez.
The European champions are said to be holding out for €80 million, and it now seems either Manchester United or PSG will help make that deal happen. AS say that both clubs have made a €60 million offer that will make Di Maria the most expensive player sold by a Spanish club, while if the deal comes off he will become Manchester United's most expensive player ever, surpassing Juan Mata.
The player is said to be angling for an exit from the Bernabeu, and while manager Carlo Ancelotti doesn't want him to leave, the club's board want to cash in. There could be a twist in the tail however, with Le10 Sport claiming that PSG are willing to go up to €70 million.
Manchester United have been linked with a £40 million swoop for Arturo Vidal and also need to strengthen their back line - which could rule them out of a bidding war for Di Maria.
Other Manchester United transfer news:
Arturo Vidal is now Manchester United's number one target, but will cost £40 million. The Red Devils could move to lessen the cost of the deal by offering winger Nani as a makeweight (The Guardian)
Manchester United are ready to cut their losses and take a hit on Wilfried Zaha. The winger signed for the club for £15 million, but he could now be on his way to Newcastle for just £7 million (Daily Mail)
At the other end of the pitch, Van Gaal remains keen on Mats Hummels as he ponders deploying a 3-4-3 system next season, and he views the German as the perfect man to fit his approach. Thomas Vermaelen is also on the club's radar. (The Guardian)
Serie A outfit Fiorentina have been credited with an interest in Marouane Fellaini (The Express)
Toni Kroos will join Real Madrid for €25 million after the World Cup - having snubbed the chance to join either Real Madrid or Chelsea (The Independent)
Manchester United are eyeing a move for Wolfsburg full-back Ricardo Rodriguez in order to provide new-boy Luke Shaw with some competition (Daily Mail)