Herrera joined Manchester United in 2014 under Louis van Gaal and has since made 177 appearances for the club and scored 19 goals.
The Spain international’s contract with United expires at the end of the season and he is free to sign a pre-contract agreement with foreign club.
United are understood to want to keep Herrera but so far the two parties have not been able to agree on an extension.
And Bilbao are ready to take advantage of the uncertainty and bring back as many as three of their former stars.
Bilbao are 17th in La Liga and are under threat of relegation.
And Alkorta says they will look into deals for Herrera, Bayern Munich’s Javi Martinez, Alaves’ Ibai and Tottenham’s Fernando Llorente.
“We are interested in Javi Martínez, Ibai, Ander Herrera, [Fernando] Llorente… we’re interested in all of them,” Alkorta told Spanish outlet Cadena Ser.
“We still haven’t made progress. We’ll get together on Tuesday to see what can be done.
“I have just returned and this week we will look at our options.”
Last week, caretaker United boss Ole Gunnar Solskjaer said he doesn’t expect any incomings this month.
“I’m happy with squad I’ve got,” Solskjaer said.
“I don’t expect anyone to come in but obviously if the club have targeted and identified a few and I think they might fit here, I’ll give my thumbs up.
“But it’s not just my decision.
“Nobody had to tell me I had loads of talented players [to inherit at United]. I can see that from outside.
“We didn’t speak too much about what’s been going on, to be honest.
“It was about seeing what you can do to go back to the way we used to play, the way Man United are supposed to play, exciting football.”