West Brom chief Ashworth: We've wanted Rosenberg for five years
Ashworth revealed Rosenberg has been on the club's radar for several years.
He said: "We've been keeping tabs on Markus for two or three years now and getting a player of his calibre is a real coup for this club.
"He had offers from Spain, Germany, Turkey, Russia and China and we're delighted he has chosen this club as the next step in his career.
"Markus is quick, has a great work ethic and can play in a couple of different positions.
"He's scored goals wherever he's played and gives us a different option to what we've already got."