Grant's future has been the subject of intense speculation with the Hammers struggling at the bottom of the Barclays Premier League.
The former Chelsea and Portsmouth manager was reported to be three games away from the sack in December and last week had to cope with reports the London club were poised to replace him with Martin O'Neill.
Moyes suggests the Israeli has been treated poorly and, while he accepts all managers must be accountable for poor results, the Everton manager is pleased he works for a low-profile chairman in Bill Kenwright.
Moyes, whose side face West Ham at Goodison Park, said: "I wouldn't like to be treated that way - but I wouldn't expect my chairman to treat me that way.
"I have got a chairman who I think would do things correctly.
"I don't know what is going on at West Ham, I can only pick up what people put out in the media or through television.
"Maybe it is a different scenario at the club but I'm just saying I think Bill and the board here do things well.
"Avram got to the Champions League final, he has managed top players, he has managed a team that didn't have any money and got them to the FA Cup final - I think you are talking about someone who has certainly got the attributes as a manager.
"Not all managers are going to win every year, so that's going to mean some are in the spotlight.
"Unfortunately it has been Avram for most of the year but you can't say he has not handled himself with complete class. Maybe the people around him haven't done that.
"He has been dignified as a manager but he has also gone about his job and tried to get the best he can.
"I'm sure it is never easy if you are picking up papers or hearing you are about to go.
"If I don't win games I'm no different to anyone else but I hope I'd be treated in a fair and dignified manner."
Grant is determined to put a turbulent week behind him despite continued doubt over his position as West Ham manager.
The club's conduct has generated considerable sympathy for Grant, but the Israeli refused to discuss his personal situation.
"What's happened has happened off the pitch and for me it belongs in the past," said Grant.
"What's important for West Ham, the fans and everybody is to concern ourselves with what happens on the pitch.
"We have problems that we need to deal with and not discuss the past. Now we have a game against Everton.
"I need to do my job in football and not think about other problems. My personal issue is not the problem.
"It's my problem, it's not the problem of everyone. It's not important to talk about what Avram feels or doesn't feel.
"I feel that I need to do my job and that I'm the manager here. We need to take points because I believe we can get out of relegation.
"My personal matter is not important. The team is more important than anybody."