Osman, who has come through the youth ranks at Goodison Park, was given his debut in the first-team by Moyes a decade ago.
Moyes' current deal is due to expire in the summer and he has said he will not open talks until the campaign has ended, which has led to speculation about his future.
But Osman believes there is a good chance that Moyes will extend his deal, given the job he has done since taking charge.
"In the 10, 11 years he has been here, he has taken us to fighting from relegation every season to the final stages of cup competitions and to the top end of the table, so his record speaks for itself," Osman told Sky Sports.
Osman, who has become an England international this season, says Moyes' style of management is one of his strongest points.
"[The best thing is] the manner in which he has done it, we will see what happens at the end of the season and hopefully he stays," he added.
"He gave me the chance and my debut and I am still in the team so I am thankful for that."