The 32-year-old has only worked under Moyes, with the exception of loan spells early in his career, and admits he is excited about the dawning of a new era at the club.
Osman last week penned a two-year extension to his Everton contract, and says he is looking forward to the challenge of impressing the new boss.
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“Then it’s about getting my head down, working and achieving something for the club next season.
“It’s going to be a new challenge and someone new to impress which should raise the bar for everyone at the club.
“Everyone will have to step up and perform, so I look forward to the challenge.”
Last season saw some of Osman’s best performances in an Everton shirt, the midfielder earning his first caps for England after, according to Moyes, carrying the Toffees for large spells of the campaign.
The England international agrees with the new Manchester United boss, and cites a move to the centre of midfield as key to the improvement in his form.
He continued: “I have played a lot more games in the middle of midfield which I always said was my favoured position and hopefully we all saw the benefits from that.
“To have played for this Club all this time and then to finally get an England cap on top of that, I couldn’t have been happier with the way it went.”