Liverpool have been working on a deal for weeks, having identified the France international as a priority target following Mohamed Salah's departure, and had already been negotiating personal terms with the player.
But French broadcaster Canal Plus reports that the Reds have now told PSG they lack the means to match the Ligue 1 champions' asking price of around £130m, a development that could open the door for Arsenal to swoop instead.
The Gunners had been focused on a different target for the left flank throughout the summer, coming close to agreeing personal terms with Vinícius Júnior before the Brazilian opted to sign a new six-year, £480,000-a-week deal to stay at Real Madrid, ending Arsenal's interest.
That left Mikel Arteta still searching for reinforcements out wide, particularly after allowing Leandro Trossard to join Besiktas, and Barcola is described by the Independent as an "attractive option" as talks over Liverpool's deal appear to stall.
Arsenal would nonetheless have to move quickly to leapfrog Liverpool, who have been negotiating with PSG for some time already.
Barcola, 23, has made 152 appearances for PSG since arriving from Lyon in 2023, scoring 39 goals and contributing to two Champions League triumphs and three Ligue 1 titles.
He came off the bench in the 83rd minute as PSG beat Arsenal in last season's Champions League final. With the French champions having strengthened their attacking options further over the summer, Barcola could be facing a reduced role at the Parc des Princes regardless of who wins the race for his signature.