The 28-year-old has signed a new contract to remain in Bremen until 2012, adding one year to his previous commitment, but will spend the rest of the season in Spain.
Rosenberg missed almost all of last season as he fell out of favour after injury, but he is eager to prove himself with the club once again.
"This is a good solution for all parties," said general manager Klaus Allofs. "Markus will get regular match practice, which despite his improving form we might not have been able to give him as we go for Champions League qualification.
"We are sure that he will succeed in Spain and demonstrate his quality again."
Rosenberg added: "With this situation, it gives me a clear goal. I'm sure I'll come back even stronger."