United opened their 2018-19 Premier League challenge with a 2-1 win over Leicester City on Friday night, and now face a trip to Brighton & Hove Albion next weekend.
The 20-time English champions will then take on Tottenham Hotspur on August 27, before fixtures against Burnley, Watford, Wolverhampton Wanderers, West Ham United and Newcastle United.
Moyes has said that his former club have the chance to put some serious points on the board in the first half of the campaign.
"I agree to the point that it doesn't matter who you're playing in the Premier League, they're tough. But if you look at the games that Manchester United play in the first nine or 10 games, you're not playing a City, an Arsenal, a Chelsea in that period," Moyes told BBC Sport.
"I wonder if Jose [Mourinho]'s words the other day, 'Let's wait until the end of November', I would be very surprised if United weren't top of the league come the end of November. I think that the fixtures that have come out for them and the way it's worked, I think will give them every opportunity to be there."
Moyes signed a six-year contract with United in May 2013, but the Scot lasted just 10 months at Old Trafford.