Koeman decided against offering Osman a new contract at Goodison Park, with the 35-year-old now searching for a new club.
However, the attacking midfielder does not appear to hold any grudges against the former Southampton boss - and he has backed him to impress on Merseyside.
“Its a good appointment,” Osman said. “If I was picking myself, I’d want a guy with Premier League experience. He’s done a good job at Southampton, and he’d have been near the top of the list. Time will tell, but he looks like he has all the attributes to succeed.
“If this was his first job in management, you’d have the utmost respect for him due to his playing career. But he’s done well as a manager too.
“The players will have a lot of respect for him, and from what it looks it’ll be a good appointment.”
Since taking charge, Koeman also allowed Steven Pienaar to leave Everton - with the Dutch manager keen to make changes.
Koeman has been tasked with guiding Everton into Europe in his first season in charge, and that will require a big improvement.
The Toffees finished in 11th place last season, with their poor Premier League performance costing former boss Roberto Martinez his job.
Osman’s future is currently uncertain, with the former England international looking for a new club.
Leeds United have been linked with a move for Osman, but it remains to be seen whether he would accept a move to the Championship.