Speculation has risen over Moyes’ future as a result of the Scot's refusal to begin talks over a new deal until the current campaign has finished.
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His current deal due to expire in the summer, but Osman is keen to see Moyes extend his stay at Goodison Park, citing his achievements since taking over at the club in March 2002.
"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.
"[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 gave me the chance and my debut and I am still in the team so I am thankful for that."