The Hammers are sitting in sixth in the league table on 45 points, but they have a game in hand over Chelsea, who are fourth and are on 47 points.
After just managing to stay up in the Premier League last season, the Hammers have sprung a surprise this season and much to the shock of many, they have continued to sustain their challenge to finish in the top four.
Moyes stressed that his side can break into the top four but conceded that there are a number of teams in the mix who were justifiably predicted to finish above his side at the start of the season.
He is delighted with the resilience his West Ham team have shown to stay the course but did admit that qualifying for the Champions League could be a level above where they are as a team at the moment.
Asked about breaking into the top four, Moyes told the BBC: “It’s a possibility.
“But we have got to say that I can see four or five teams who are easily chasing us down and four or five teams, who at the start of the season every one of your pundits will tell you that they were going to finish above West Ham, me too.
“Those teams are still there and they are still going to push hard us and challenge.
“But we are putting on a really good fight at the moment, we have been resilient in most of the games. We’ve had a go.
“Yes, we have lost some and some we have won, but overall we need to keep trying to stay up there.
“Whether we can finish fourth is probably above our station at this point in time.”
West Ham will look to get back to winning ways when they host Leeds United at the London Stadium on Monday night.
