Herrera was the lynchpin of Bilbao's midfield last season as they finished fourth in La Liga and was believed to have a £28m buy-out clause in his contract, which rises to £32m after July 1.
He scored five goals last season, but despite being one of the most impressive performers in domestic football, is yet to break into the Spain team.
An Athletic statement read: "Athletic want to announce that they have this week rejected an offer from Manchester United, of around 36m euros (£28.85million), for the transfer of the economic and federative rights of the player Ander Herrera."
Despite the rejection Sky Sports News reporter James Cooper insists Manchester United are 'relaxed about the situation'.
He said: "They have been negotiating with Athletic Bilbao for the past few months. This is the way they do things in the Basque country. We saw it with Javier Martinez going to Bayern Munich a couple of seasons ago.
"But what Athletic Bilbao don't like doing is letting other people know they're happy to let their big players go to other clubs, so there will be a little bit of posturing here while this plays out."
Last summer former United boss David Moyes reportedly targeted the 24-year-old with bizarre scenes in Spain on deadline day when three men claiming to represent United tried to negotiate the deal at the Spanish FA headquarters.
The negotiations broke down, with United claiming they knew nothing about the three men.
However, new manager Louis van Gaal has made a firm move for the player as he looks to rebuild the squad after last season's poor campaign.
Southampton and England left-back Luke Shaw could be the next target, with United having reportedly increased their offer for him to around £34m.
United had also been linked with a move for Ezequiel Garay, but the Argentine defender has this week agreed to join Russian club Zenit St Petersburg from Benfica.