Valencia-based Gols Media claim Los Che are intent on including the Dutch international in their summer clearout this summer and he could be available in the region of €10m-12m.
The exits of Valencia club captain Dani Parejo and Francis Coquelin to Villarreal from Los Che were also confirmed last month, following the exit of star winger Ferran Torres to Manchester City and striker Rodrigo Moreno has joined Leeds.
Cillessen, 31, spent five seasons at Ajax between 2011 and 2016 before a €13m move to the Camp Nou, but he played second fiddle to Marc-Andre Ter Stegen and transferred to the Mestalla last summer.
Cillessen moved to Los Che in a €35m deal while Neto went the other way for €26m plus €9m in variables and has featured in 24 La Liga games last campaign.