Athletic announced earlier on Wednesday that they had rejected a £28.85m bid from United who unsuccessfully tried to sign Herrera on deadline day last summer.
Herrera is thought to have a £28m buy-out clause in his contract, which rises to £32m after July 1.
He scored five goals last season as Athletic finished fourth in La Liga but 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 of their latest offer, 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."
"This is the way they do things in the Basque country." No, it's just that they're waiting for July 1. 5 more days = + 4m. But it also looks like they do things the reverse in Basque country. Take a medical, sign a contract, then eventually get legal permission to do those things for another club and pay the transfer fee.- dr soccer