Iker Muniain's goal in the second half proved to be the difference between the two teams in their La Liga match at San Mames.
"We lost a great opportunity to enter the Champions League places," Valverde told reporters. "Bilbao pushed us, but we saw that we could harm them.
"We're not very happy because we know we should have got more from this match. We thought we could get something here today."
Valencia are currently sixth in the league table with 42 points from 27 matches.