The Gunners are looking at potential options should Arsene Wenger’s time in charge at the Emirates come to an end.
Express Sport understands there are split views behind the scenes at Arsenal over his future.
Crystal Palace’s 3-0 win over the Gunners on Monday has left Champions League qualification unlikely.
And Wenger’s contract has still not been sorted ahead of next season, meaning alternatives are being identified.
Ernesto Valverde has been in charge at Athletic Bilbao since 2013 and impressed during that spell Next Arsenal manager odds: Who could replace Arsene Wenger? Tue, April 11, 2017
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Arsenal wanted to know if he would be willing to move to the Premier League
According to Spanish newspaper Marca, a representative of Arsenal has given 53-year-old Valverde a phone call.
The Spaniard is currently in his second stint as Bilbao manager, but has confirmed to the club he is set to leave.
Valverde is among the top candidates for the Barcelona job too with Luis Enrique stepping down this summer.
However, with his replacement at the Nou Camp not announced yet, other teams are looking at Valverde.
Arsenal wanted to know if he would be willing to move to the Premier League.
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Arsene Wenger's future at Arsenal is still unknown
And joining the Gunners would apparently be an enticing proposition for Valverde, who has always liked the idea of working in England.
But Barca have made their interest known, despite publicly denying an approach.
Marca add the La Liga giants will need to move quickly if they want to hire Valverde, especially with other teams in the hunt.
Still no announcement has been made over Arsene Wenger yet at Arsenal, who have also been linked with Max Allegri and Thomas Tuchel.