The La Liga champions have been heavily linked with a move for their former forward since the end of last season.
However, Ernesto Valverde's side have now given up their pursuit of the Brazil international ahead of Monday's European transfer deadline.
Barcelona were keen to bring the 27-year-old back to the Camp Nou but ultimately concluded that they could not put the club's financial health at risk.
The Catalans had been working on a players-plus-cash proposal, but PSG insisted they wanted to recoup the £198m they paid for Neymar in 2017.
However, Barcelona have not ruled out returning with another offer next summer.
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