Everton will host Premier League title hopefuls Manchester City at Goodison Park on Wednesday night.
Since Moyes left Goodison Park for Old Trafford in 2013, Everton have had four permanent managers, including summer appointment Marco Silva.
On the other hand, United have been unable to replace their serial title winner, with three permanent managers sitting in the hot seat since his retirement six years ago.
In recent weeks, Silva's struggles have become apparent, none more so than their FA Cup defeat to Championship side Millwall last month [BBC Sport].
But Gray insisted that they must keep hold of their under-fire manager and be patent with him like the previous regime was with Moyes when they first appointed him.
"I just don't want to be repeating the same mistakes of 'getting another guy in, getting another guy in'," Gray told beIN SPORTS.
"Somebody at some time has to be given time. Like David Moyes was when Bill Kenrwright appointed him and gave him 10 very stable years for Everton.
"Since Moyes left, on a smaller scale, it's a bit like Manchester United trying to find a replacement for Fergie. Everton have not been able to find the replacement for David Moyes."
At this moment in time, Everton are sitting outside of the top 10 and they can slip further away if they don't pick up positive results.
They host second-placed Manchester City on Wednesday night at Goodison Park, with a win for the away team ensuring they leapfrog leaders Liverpool in the table.
It is a tough ask for Silva's side to take all three points, but the supporters know with the likes of Richarlison and Gylfi Sigurdsson in their ranks, they have a chance of causing an upset.
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