The Hammers issued a statement on Tuesday refuting claims Moyes was set to be replaced as manager at the end of the season.
Asked about Moyes' situation at the club, Nolan told talkSPORT: "I thought David Moyes did fantastically well to turn it around and I think he will still keep them up, I think they've got enough quality.
"It's one of those things, when you move to a new stadium it's so tough and it takes time to bed in. It happened to Arsenal when they moved from Highbury to the Emirates and it's the same now with Tottenham at Wembley.
"At the minute I feel the fans are a bit discontent because they don't feel like it's home, but the team are fighting every week to try and do something.
"I speak to them most weeks. I really believe they'll be ok and that David Moyes is the right man for the job at the minute."