Asked after Barca's 2-0 win over Getafe on Sunday if he had thought about resigning, Valverde said: "Well, no."
Arturo Vidal grabbed a rare goal for the hosts before a late Mauro Arambarri own goal secured three points for the Blaugrana.
Despite sealing another domestic victory, Barca fans targeted their players at Camp Nou, with Philippe Coutinho and Sergio Busquets among those subjected to whistles.
Valverde, who agreed to a contract extension until the end of 2019-20 in February, was unsurprised by the crowd's response and said it was only natural after their Champions League exit.
"I expected that at the beginning they would whistle. How are they not going to whistle us? It is normal," he said.
"But I knew that they would then position themselves with us.
"And there seems to be noise but people are from your team and I'm grateful."
Barca wrap up their La Liga campaign at Eibar on Sunday, before taking on Valencia in the Copa del Rey final on May 25.