Osman hopes Everton can win a trophy as a reward for the efforts of Moyes during his 11 years at Goodison Park.
He said: "The manager has been a fantastic servant to the club. If he was to ever leave, I'd be disappointed.
"But you don't always get what you deserve. You don't just get to win a cup because you've done well and deserve it.
"It's up us as a group of players, manager and staff included, to try and turn what is deserved into actually doing something."