Herrera was the subject of a bid from the Premier League champions during the week, which was rejected by the Basque club.
Herrera had started both of Athletic's league games this season - against Real Valladolid and Osasuna - but played no part in the early Primera Division kick-off at the Bernabeu, with Mikel Rico starting instead.
But speaking after the game, Valverde was adamant the bid from David Moyes' side was not related to his decision to leave Herrera out of his team.
"It had nothing to do with the bid from Manchester United, I left him out because I chose the players I thought were best suited to the game," Valverde told a press conference.
"I'm not thinking about speaking to him to convince him about anything and I'm not worried about anything else.
"I haven't spoken with the player but what I'm going to repeat is that I don't expect the club to make any moves in the transfer market tomorrow (Monday).
"I do not know what will happen. I want Ander to stay with us, and that is what I expect will happen."
Asked why he thought Herrera would stay, he responded: "It would be harder to explain why I thought he was going to leave."
Before the match, Valverde declared during a pre-match press conference that Herrera was fit to play against Madrid.
The 24-year-old progressed through Real Zaragoza's youth ranks before joining Bilbao in 2011 for 8million euros, and was part of the side that defeated Manchester United 5-3 on aggregate in the last 16 of the Europa League in 2011/12.
He has a buy-out clause in his contract which is widely believed to be 36million euros.