Moyes' position is under scrutiny with his side heading to Leeds on Wednesday night bidding to avoid a sixth consecutive Premier League defeat.
The Hammers have slipped to 17th in the table - level on points with third-bottom Nottingham Forest and only two points clear of last place.
But speaking ahead of West Ham's trip to Elland Road, manager Moyes is confident he is not facing the sack.
"I am feeling really good because I have great support from [co-owner] David Sullivan and the board," he said.
"They have been right behind me which is really good because you can put yourself under pressure a little bit. But overall they have been really good and really supportive in backing me."
West Ham suffered a 2-0 defeat at home to Brentford last time out - four days on from a 3-1 loss at Arsenal - with some fans questioning if Moyes remains the right person for the job.
However, the 59-year-old Scot, in his second spell in charge of the London club, said: "I think the West Ham supporters have been incredibly supportive.
"They have seen what has gone on here in the past three years, and would probably say the opposite [to the club changing manager].
"They are desperate for us to do well, and desperate for the manager to do well.
"We have to buck up, do better, play better and find a way of winning games which is they key.
"I have been in this position twice before, and we did a very good job, but I don't think we are in that position now because we have a better team, so we just need to show that form."
Moyes believes summer signing Gianluca Scamacca will be able to help turn the club's fortunes round.
The £30million arrival from Genoa has scored just twice in the Premier League, and is without a goal since the beginning of October.
"We can and we will [get more out of him in 2023]," said Moyes of the 24-year-old. "It takes players time to settle into the Premier League.
"He is a young centre-forward who has come from Italy. He has a big future, we like him a lot, and he has so many good attributes, but it can take time.
"He has had a couple of injuries, and was behind when he joined us, so it might just take him a little bit longer to get used to it."
Striker Michail Antonio could be available to face Leeds despite suffering with illness and Morocco international Nayef Aguerd is in contention after overcoming a virus, but fellow defender Kurt Zouma is absent.