The veteran, who has struggled with injuries this season, will be out of contract as a player this summer.
However, the experienced striker could sign a new deal "in a technical capacity".
"We've offered him the opportunity to stay with the club, though logically, not as a player. We know he has a lot of thinking to do, but we can't rule out the possibility of him staying with us in a technical capacity," Mourino is quoted as saying by Inside Spanish Football.
"He's earned the affection of our fans and has also been a great example of professionalism."
Bermejo made just one league start in the 2013-14 season.