The Premier League club had hoped to agree to a deal with Real Madrid, but the emergence of the Ligue 1 side could see United outmuscled in transfer talks.
The Old Trafford side had been confident of reaching an agreement with Real Madrid to sign the winger this summer as Louis van Gaal continues his squad overhaul.
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And PSG president Nasser Al-Khelaifi has reportedly held talks in Madrid with the Spanish giant.
The offer of Champions League soccer from the French capital club is also understood to appeal to Di Maria, despite the presence of Van Gaal at Old Trafford.
United has pressed ahead with its summer transfer plans during the World Cup despite the Dutchman's continued involvement with the Netherlands in Brazil.
Ander Herrera was signed from Athletic Bilbao, while Southampton teenager Luke Shaw made a big-money move from Southampton.
But Van Gaal could miss out on another primary transfer target after Toni Kroos previously turned down the chance to leave Bayern Munich for the Premier League.