President Josu Urrutia confirmed on Friday that the La Liga club flatly turned down the approach, but refused to confirm the identity of the bidding club.
United manager David Moyes is said to have turned his attention to Herrera after unsuccessful bids for Cesc Fabregas and Marouane Fellaini.
Moyes is keen to strengthen his midfield, but Urrutia insisted Herrera is not for sale.
"There is an offer for Ander Herrera,” he confirmed. "We received it yesterday and we have informed them that we will not negotiate for our players.
"Our club is different, based on sentiment. Our goal is not to make money.
"For a player to go first the player has to express his desire to leave the club.
"Herrera has not told us he wants to go, and we have said we do not negotiate for our players."
The 24-year-old signed for Athletic in 2011 and has made 63 Liga appearances for the Basque club.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.