Barca president Joan Laporta said on Friday he was not prepared "to put the club at risk" by renewing Messi's contract, as the club face up to huge financial problems ahead of the start of La Liga on August 15.
Laporta said Barcelona were no longer negotiating with Messi and the 34-year-old forward was considering other offers.
"The player has other offers and there is a time limit because La Liga starts soon and he needs time to assess his other options," Laporta admitted.
Broadcaster RMC Sport reported that Messi has informed PSG of his desire to join the club and that discussions are ongoing over a two-year deal with the option of a third season.
Le Parisien reported that Messi's father and representative Jorge had reached out to PSG after it became clear the player would not be staying in Catalonia.
"Our president is looking at ways of improving the team," PSG coach Mauricio Pochettino, Messi's fellow Argentinian, said earlier.
Manchester City, who attempted to sign him last year, had also been linked with a move. However City manager Pep Guardiola, Messi's former coach at the Camp Nou, dismissed that, saying that "right now it is not in our thoughts, absolutely not."
Messi had reached agreement with Barcelona to sign a five-year contract worth 50 per cent less per year.
But La Liga's financial fair play rules, which are expected to reduce the amount Barca can spend on salaries this season by some $235 million, led Laporta to conclude they were unable to complete the deal.
"We could not accommodate Messi's contract," said Laporta, who was re-elected as club president in March.
"The situation we have inherited is abominable and the sports salary bill represents 110 per cent of the club's income.
"La Liga's rules set limitations and we have no margin. We have known the situation since we arrived but the numbers we have seen are even worse than we thought."
Barcelona have total debts of almost 1.2 billion euros, with more than half of that needing to be repaid in the short term.
Laporta said they found the club's losses for the season just ended were almost 500 million euros, more than double what they expected.
La Liga announced a deal with US private equity firm CVC earlier this week, under which it agreed to sell 10 per cent of the business for 50 years, in exchange for an immediate injection of 2.7 billion euros. Clubs in Spain's top two divisions would receive 90 per cent of the funds.
Laporta suggested that with the new salary cap rules, Barca could only have registered Messi if they agreed to the CVC deal but insisted it was a risk he was not prepared to take.
Barcelona and Real Madrid have publicly objected to the CVC agreement. "I am not willing to mortgage the club for anyone. We have an institution that is above everyone, even the best player in the world," Laporta said.
Messi tried to leave Barcelona for free last year but was forced to stay after the club told him an exit clause in his contract had expired.
He went on to score 38 goals in 47 games in all competitions. That, and Laporta's re-election as president, changed the dynamic and the expectation was Messi would sign a new deal.
"Leo wanted to stay and he's not happy," said Laporta. "I have told the players that a new stage in Barca's history is beginning and that we have to continue winning without Messi."
Laporta said he felt "sadness and disappointment" but that he "did everything possible" to keep Messi, who joined Barcelona aged 13 and has 672 goals in 778 appearances for the club.
Teammate Sergio Busquets posted a farewell message to Messi on Instagram, writing: "You arrived as a child and leave as the best player in history."
With Barcelona, Messi won La Liga 10 times and the Champions League four times. He is also a six-time winner of the Ballon d'Or.
"Leo Messi has been an icon and has left some historic moments. We have eternal gratitude to him," Laporta said.