Valverde is under pressure at Barca after their stunning Champions League collapse at Liverpool this week.
And it's emerged that his new deal, which runs to 2020, includes a break clause that both parties can trigger at the end of the season. Valverde only penned terms earlier this year.
Meanwhile, El Chiringuito says former Barca captain Koeman is under consideration to succeed Valverde.
The former Southampton and Everton manager's appointment would also assist Barca in their attempt to sign Ajax defender Matthijs de Ligt. Frenkie de Jong is already leaving Ajax for Barca this summer. Both players are key members of Koeman's Oranje squad.