Herrera joined Athletic from Real Zaragoza in 2011, and made 128 appearances for the Basque outfit in all competitions before joining United in the summer of 2014.
The Spaniard has made 161 appearances for the Red Devils since his arrival, and was presented with the Sir Matt Busby Player of the Year award for his performances during the 2016-17 campaign.
Herrera struggled to hold down a regular spot in the first team last season, however, and his future has been called into question.
United recently decided to activate a one-year extension clause in the 28-year-old's contract, but according to AS, Athletic are desperate to re-sign him in this summer's transfer window.
The report claims that the Basque side want to make Herrera their highest-paid player, although they will find it difficult to tie up a deal until the 20-time English champions sign a replacement.
Herrera only started 13 of United's 38 Premier League matches during the 2017-18 campaign.