The news of PSG's interest was first reported by L'Equipe.
Rashford, 24, has one year left on his contract at United with an option for another season that the club alone can trigger, meaning in effect, he has two years remaining.
PSG are yet to start discussions with Manchester United, sources have told ESPN. And United could ask for a high fee if they agree to let him leave. However, the Rashford camp has been receptive to the Parisians' approach so far.
Rashford's representatives met with PSG last week, sources added to ESPN. The French champions said they would be prepared to offer Rashford a large contract though didn't discuss what transfer fee they'd be willing to pay United.
New manager Christophe Galtier said at a news conference Thursday that he needed another forward, to add to Kylian Mbappe, Lionel Messi, Neymar, Hugo Ekitike and Pablo Sarabia, and that the club were working on a player who could play wide or as a No. 9.
"We wish the arrival of a new striker," Galtier said. "With a very busy calendar until Nov. 13, plus the World Cup, obviously we wish this arrival. The club is working very hard.
"I am in direct contact with [sporting director] Luis Campos on this and he is in direct contact with the management and our president. You mustn't make mistakes, you mustn't do anything, and then there are things that can be done and things that can't. Will this player make it? We have no guarantee on that. I know that the club will not take and pile up a player, it must be a real added value, and the club is working in this direction but whether by the end of the transfer window an offensive player will arrive I cannot confirm it but the club is working hard on this position."
"This player" is Rashford. Mauricio Pochettino was keen to bring Rashford to Paris when he was in charge at the Parc des Princes before being sacked earlier this summer. New sporting director Campos is also an admirer of the England international.
For Rashford, who has been struggling at Old Trafford for the last 18 months, PSG represent the possibility of an intriguing new challenge. He scored only four goals in 25 appearances (13 starts) in the Premier League and one in five Champions League games last season.
He also lost his place in the England squad, and three months before the World Cup, he faces an uphill battle to regain his place in Gareth Southgate's plans.
ESPN correspondent Rob Dawson contributed to this report.