The London-based club began their 2020/21 Premier League campaign with a 2-0 home defeat to Steve Bruce’s Newcastle United side last weekend.
West Ham now have a tricky schedule ahead, with Moyes’ side set to take on Arsenal, Tottenham, Liverpool and Manchester City in their next six league games.
However, Irons star Snodgrass is positive that the team can take some points off the Premier League big guns, having taken points off Chelsea and Manchester United towards the end of last term.
The Scotland international went on to reveal that manager Moyes has challenged the West Ham team to behave and perform like a top ten club, which he feels they are capable of.
“The manager has challenged the players because of how well we finished last season, and why not?” Snodgrass said on West Ham TV.
“You need to demand high standards of yourself and your team-mates.
“The manager has been here for probably eight or nine months and he is trying to get his ideas across and getting the lads thinking like a top-half-of-the-table team and we’ve definitely got the ability, but it’s just about the belief.
“It’s about starting the season well.
“We’ve got some tough games coming up, but the lads finished last season well and got some great results against some great sides, so why not?”
West Ham bounced back from their weekend defeat to Newcastle by beating Charlton Athletic in the EFL Cup on Tuesday and will be looking to earn their first points of the season against Arsenal on Saturday.