Barca paid an initial €13 million (£12m/$15m) to sign Cillessen from Ajax in 2016 and he made just four appearances in La Liga and the Champions League over his first two seasons, with his main usage being in the Copa del Rey.
The 29-year-old has started all three of the Catalan giants' International Champions Cup fixtures but was replaced by Marc-Andre ter Stegen, who only returned from a post-World Cup break this week, at half-time in Saturday's 1-0 loss to AC Milan in Santa Clara.
When asked in the mixed zone after the match if he will move on before the end of the transfer window, Cillessen simply responded: "I don't know."
But head coach Valverde made it clear he wants the Dutchman to remain at Camp Nou and provide competition for Ter Stegen.
"My idea is to have two good goalkeepers, the two we have now, Marc and Jasper," said Valverde.
"It's true that [Cillessen has] not played as much as Marc, but that's what being a goalkeeper can be like.
"He knows what I think. I don't think he will be hard work if he stays – we are very happy with his professionalism and his attitude. We hope he stays with us and is happy here."