Martinez is expected to soon add to his new squad with free agent Antolin Alcaraz being lined up.
Osman told evertonfc.com: "I have met the new manager. He seems a really nice guy and from what I have seen he is already working as hard as he can to make this great club better.
"It's just known that when a manager leaves he takes his backroom staff with him, so its not just one face that leaves its up to six or seven. On the other side of it, a new manager brings in new faces with him.
"There has been a summer of big changes at the club but I don't think it will change what we are and we are made up of and the spirit that we have had here. We have had that over all these years and I don't think that will change."