Having failed to secure either Thiago or Cesc Fabregas this summer, United have identified a third Liga target in the deep-lying midfielder, who has represented Spain at youth level.
However, an opening bid has been turned down, with Bilbao determined to resist any approaches for the player.
"There's an offer for Ander Herrera," chairman Josu Urrutia told reporters.
"Our club is different, it bases itself on sentiment. Our goal is not to make money. We received the proposal last night and we said that we don't negotiate for our players.
"For a player to leave, he first has to show that he wants to leave the club then deposit the clause."
Herrera has a release clause of 36m euros (£31m) and a bid of that value would leave Athletic powerless to prevent his departure.
The 24-year-old joined Athletic from Zaragoza in 2011 and impressed in the two-legged victory over United in the Europa League in March 2012.