Carlo Ancelotti's side have moved first to try to beat Manchester United to the punch, but the report claims Sir Alex Ferguson has instructed United club officials to contact Ronaldo's agent in response.
United are keen to ward off interest from PSG as they look to bring back the player they sold for £80m in 2009. And the Premier League champions have held discussions with sponsorship giants Nike to attempt to put an attractive package together to bring Ronaldo back.
The report claims United are prepared to offer up to £65m plus a player if Ronaldo's representatives make it clear he's willing to return to Old Trafford. The most likely United player to be used as a makeweight is Nani.
But Ferguson's side have been blindsided by this move from PSG. The Sun quotes a United source as saying: "United are not happy now PSG have made their move with a substantial offer.
"They will not give up, however, and have told Ronaldo's agent of their intent."
The report claims PSG have offered Real Madrid £80m for Ronaldo, and would be able to better any wage package put together by United.
The Old Trafford side's highest earner is currently Wayne Rooney on £250,000-a-week. But a return to Manchester is thought to be Ronaldo's preferred destination, and the financial package may not be the main motivator.
The Portugese forward enjoyed a hugely successful stint under Ferguson, developing from an 18-year-old prodigiously talented winger to a world class Ballon d'Or winning goal machine over a six-year spell.
He has two years remaining on his Real Madrid contract, and has so far failed to put pen to paper on an extension.