The 25-year-old striker spent seven years working under Moyes at Everton. Moyes handed Anichebe his Premier League debut in 2006.
“David Moyes is a good guy and a great coach,” Anichebe told the Daily Mail.
“People say the same about Marouane Fellaini and ask whether they are ‘Manchester United people’.
“I’m not sure what that means. They are successful. Full stop.
“They will do a good and honest job. I don’t know if he will be successful in the future, but they have been so far. Why not?”
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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.