Last month, ESPN reported that Messi was unlikely to sign a new contract with the club, and sources have now said that both parties have agreed not to extend his stay with the French champions.
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After PSG's 3-1 defeat to Lorient -- their third loss in their past four league games -- Messi flew to Saudi Arabia without asking again for permission, sources told ESPN, meaning he could not attend training Monday.
The incident further complicated the relationship between the player and the club. Sources told ESPN the situation created tension within the dressing room, with some players unhappy about the incident.
Sources said that an agreement in principle was found between PSG and Messi before the World Cup for him to sign a new deal in Paris, but the situation has since changed.
On Wednesday, several hundred PSG supporters gathered outside the club's Paris headquarters to express their anger at Messi, as well as teammate Neymar and president Nasser al-Khelaifi. A small group of supporters later protested outside Neymar's home, calling on him to leave the club.
"Paris Saint-Germain most strongly condemns the intolerable and insulting actions of a small group of individuals that took place on Wednesday", the Ligue 1 club said in a statement.
"Whatever the differences of opinion, nothing justifies such actions. The Club gives its full support to its players, its staff and all those targetted by such shameful behaviour."
Messi, 35, has failed to replicate his excellent World Cup form with PSG and the club doubt whether Messi can still deliver in big Champions League matches.
Sources told ESPN that there is an offer from Saudi Arabia for Messi and he is aware of interest from former club Barcelona and Major League Soccer club Inter Miami.
Barcelona are looking into the possibility of re-signing Messi, although LaLiga president Javier Tebas warned that the league won't change its financial fair play rules to allow him to return.