Moyes' current deal runs to the end of next season and Everton are eager to extend and improve the manager's terms.
Speaking courtesy of BetSelect, Gray told the Liverpool Echo: “When you look at what he did at West Ham, where he delivered the European trophy and then got finished? Come on.
“Then look where West Ham are now, three managers later. Yeah, it’s extraordinary, the people make these decisions.
“I was delighted West Ham got rid of him because I knew he was coming to Everton, and it was a great decision. It was a no-brainer for Everton to bring David back into the new stadium as well.
“He should have been the man who led Everton into the new stadium, and he was. I think we can see improvement.
“Yes, he may be a bit pragmatic in his coaching at times. That’s fine. That will develop, as he showed us when he worked with players like Declan Rice and Jarrod Bowen on his team and won European trophies.
“Of course, he can play the type of football that we want to see, and we can see evidence of improvement this year compared to last year, and it's right on the verge of European football.
I would like to think there’s no argument for him to get a new contract, no argument at all for me.
“Give it to him, absolutely. I would be amazed if Everton doesn’t offer David a new contract and I’d be disappointed as well.”