The Costa Rica international was one of the revelations of the 2013-14 La Liga campaign and also impressed at the World Cup with a string of fine displays.
Navas had also attracted the interest of Monaco with his sublime performances, but he seems destined to continue his career at Bayern instead.
"He's going to one of the best teams in Europe and I'm sure that he's going to enjoy playing there and will be good competition for his new team-mate, [Manuel] Neuer," Perez was quoted as saying by Teletica Deportes.
"We're happy for him, because I think he deserves it as a player and a person. He's earned it. He had a great year and we are happy to have helped him grow in his career."
The 27-year-old started his professional career with Saprissa and also wore the Albacete jersey before joining Levante in the summer of 2011, where he developed into a key player for the Spanish club.