The Paris Saint-Germain winger, 23, informed the French champions on Sunday that he does not intend to extend his contract, which runs until June 2028, opening the door to a potential exit this summer.
Barcola has since held talks with several Liverpool executives as well as Iraola personally, and is understood to be eager for the two clubs to reach an agreement that would allow him to test himself in the Premier League. Liverpool have begun discussions with their Parisian counterparts, but the two sides remain far apart, with PSG reported to be holding out for a fee in excess of £140m, a figure that would make Barcola the most expensive signing in Premier League history, while remaining firm on their financial demands despite now being open in principle to a sale.
PSG's valuation is understood to reflect Barcola's importance to the club over the past three seasons, during which he won back-to-back Champions League titles, as well as the scale of other deals completed elsewhere this summer, including Chelsea's £117m signing of Morgan Rogers and Manchester City's £116m capture of Elliot Anderson.
Liverpool are reportedly exploring ways to fund a move, including the possible sale of Cody Gakpo, who has attracted interest from elsewhere in the Premier League, as they look to add further attacking reinforcements following Iraola's appointment.
Barcola's willingness to commit personally to the move could prove significant in accelerating a deal that, as things stand, still has a considerable gap to close between the two clubs' positions. His preference for Anfield over other suitors gives Liverpool a clear advantage in any negotiation, even if PSG's asking price means the transfer remains far from complete.
Timing has also complicated matters, with Barcola's extended involvement in this summer's World Cup delaying serious discussions between the clubs until recent weeks, and PSG head coach Luis Enrique said to want only fully committed players in his squad regardless of how the saga is resolved.