PSG host relegation battling Le Havre on Saturday
Valverde's contract expires in June, like that of current Barcelona coach Luis Enrique, and most expect Valverde to replace the Asturian, who in March announced he was leaving at the end of the season.
Barcelona president Josep Maria Bartomeu will reveal the board's decision over a new coach next Monday.
Former Barcelona player Valverde, 53, steered Athletic to eighth place in La Liga this season, but has decided to call time on his second managerial stint at the San Mames.
Valverde has coached Bilbao since 2013, previously taking the reins between 2003 and 2005, before being appointed at Barcelona's city rivals Espanyol (2006-08) as well as Olympiacos twice, Villarreal and Valencia.