The Spanish playmaker was expected to complete a move to Old Trafford this week, having already agreed personal terms with the Red Devils, but United must still agree a fee for a player the Basque club remain keen to retain.
A statment read: "The Athletic Club wishes to report that we have this week rejected an offer from Manchester United of €36 million for the transfer rights of Ander Herrera."
Herrera was given a tour of United's training ground on Wednesday, having said his goodbyes to his Athletic team-mates.
However, the two clubs have yet to settle on a price for the 24-year-old, though United remain confident of eventually acquiring his signature, despite Athletic's famously tough negotiating skills.
United have been chasing Herrera for over a year, with David Moyes exploring the possibility of signing the midfielder last summer. However, that pursuit ended in farce, with imposters posing as United intermediaries attempting to negotiate with the Spanish club on transfer deadline day.