United are back in for Herrera having failed to sign the 24-year-old last summer but have seen a fresh bid for the player rebuffed by Athletic this week.
An Athletic statement read: "Athletic want to announce that they have this week rejected an offer from Manchester United, of around 36million euros (£28.85million), for the transfer of the economic and federative rights of the player Ander Herrera."
United have pushed to bring in Herrera before his transfer fee rises at the end of this month due to the terms of his contract and a deal had been expected to be completed this week.
That would have made Herrera United's first major signing since they confirmed Dutchman Louis van Gaal - currently in charge of Holland at the World Cup - as their new manager, but for the second time in 12 months the Red Devils have not found the transfer straightforward.
United had attempted to bring in the former Spain Under-21 international last August but their attempts descended into farce on transfer deadline day.
Reports emerged of three people, said to be representing United, managing to get into Spanish league headquarters for discussions about the proposed switch and then leaving an hour later, apparently unable to complete the paperwork required to finalise the deal.
It subsequently transpired that United, who dealt directly with Athletic, knew nothing of the trio's existence. Negotiations between the two clubs collapsed anyway, with then United manager David Moyes believing the buy-out clause in Herrera's contract was too high.
United have now returned for the former Real Zaragoza player, who joined Athletic in 2011, but have been unable to persuade Athletic to part with Herrera yet.