The 23-year-old winger, who has two years left on his current deal, has grown increasingly unhappy with a bit-part role under Luis Enrique despite PSG's back-to-back Champions League triumphs, with pundits in France noting that the decision to reject a new contract stems from footballing pride rather than any dispute over wages.
Barcola has started only intermittently for PSG as he competes with Désiré Doué and Khvicha Kvaratskhelia for a starting spot.
Liverpool are understood to hold the upper hand in the race for his signature as they continue their search for a replacement for Mohamed Salah, though Arsenal have also been strongly linked and remain in the picture.
PSG are reportedly prepared to let Barcola leave this summer if their valuation, thought to be in excess of €135 million, is met, which would make him one of the most expensive transfers in the sport's history.
Bayern Munich and Chelsea have also registered an interest, though Barcola is understood to have already turned down the chance to join Chelsea. Reports in France suggest the sporting project being built by new Liverpool manager Andoni Iraola is a significant factor in his growing preference for a move to Anfield over other suitors.
Whether PSG choose to sell this summer remains uncertain, with the club also working on other incoming attacking business, including interest in RB Leipzig's Yan Diomandé. Barcola's future is likely to become clearer as the transfer window enters its final weeks, with Liverpool's need for reinforcements following Salah's departure adding urgency to their pursuit.
Barcola joined PSG from Lyon in 2023 and has won two Champions League titles during his three years in the French capital, but has never fully established himself as a first-choice starter, often used as an impact substitute in the biggest matches, including both of PSG's Champions League final wins.
He impressed for France at this summer's World Cup, scoring three goals and providing an assist in eight appearances as Les Bleus reached the semi-finals, a run of form that has only strengthened his belief that he deserves a more prominent role next season.