French outlet L'Equipe were first to report the story. Negotiations between the two parties are at a standstill, casting major doubts over the World Cup winner's future in Paris, according to sources.
The French champions want Messi to take a 25% pay cut from his existing salary, which is worth €40 million gross a year, sources have told ESPN.
A new meeting between the club and Messi's father and agent, Jorge, is still scheduled for later this month to find a solution.
Sources have told ESPN that Messi was likely to sign a new deal before the World Cup but the situation has changed since then.
Messi has failed to replicate his excellent World Cup form with PSG and the club doubts whether Messi can still deliver in big Champions League matches.
Sources told ESPN that Messi, 35, is not prepared to reduce his wages and was upset after his name was jeered by a section of the club's supporters in the previous two home matches.
ESPN sources added that Messi is leaving all doors open for other teams to make him an offer.
His father had already discussed the possibility of a Barcelona return, but financially it will be difficult to achieve while the player's camp is also awaiting a proposition from Inter Miami and Major League Soccer.
According to sources, Messi is open to playing in the U.S. before his career comes to an end while there remains interest from Saudi Arabia club Al Ittihad.