The move for the midfielder was completed after a buy-out clause in Herrera's contract was triggered, enabling him to leave Atletic Bilbao after three seasons.
And the 24-year-old, who signed a four-year deal at Old Trafford with the option for a further year, is eagerly anticipating his debut under van Gaal, who will take over as manager after the World Cup.
"Louis van Gaal is one of the best managers in the world," Herrera told MUTV.
"I want to learn a lot of things and I can't wait to play for Manchester United."
Herrera also revealed fellow Spaniards Juan Mata and David de Gea had provided him with positive feedback on the club, having been heavily linked with the Premier League club 12 months ago.
Asked about the discussions he had with Mata and De Gea, Herrera added: "They told me this is the biggest club in the world."