The Basque outfit have enjoyed an impressive season in La Liga and currently sit fourth in the standings after 22 matches.
"Top four would be great but I don't want to put a roof on things," Herrera told Mundo Deportivo.
"I still think the dream is to play in the Champions League. I also think we have the capacity and potential to do it but we can't forget where we came from.
"We had a very difficult, tumultuous year in which we didn't live up to our potential. We are privileged to be how we are now but we cannot forget we came from the mud."
Herrera, 24, was strongly linked with a move to Manchester United in the summer, but remained at San Mames after the Premier League champions failed to meet the Spaniard's release clause.