The champions will open up their Premier League defence at Swansea, where they're 8/13 to win, before hosting Jose Mourinho's Chelsea, followed by a visit to Liverpool amid a rigorous introduction to life at the top for Moyes, who replaced Sir Alex Ferguson earlier this summer.
The 20-time title winners will then welcome newly promoted Crystal Palace to Old Trafford before continuing their domestic campaign away to City, who are 23/10 to regain the trophy they won in 2011-12.
Despite the difficult start, United remain 11/5 favourites to win the title ahead of Chelsea (9/4), with City having been pushed out to 23/10 third favourites since Mourinho's return to Stamford Bridge was confirmed earlier this month.
Ironically, when the Portuguese first came to the Premier League he was drawn against United in the second game of the season, albeit at Stamford Bridge, where he guided his 2004-05 team to a 1-0 win.
Sky Bet's Chris Spicer said: "David Moyes is set to get a dose of what it's like to be expected to win every game now that he's boss of Manchester United and the results of his opening five fixtures could set the tone for the season.
"However, everyone has to play each other home and away at some point so we're sticking with our decision to make the champions favourites to retain their title.
"We expect the United board to keep faith in the man Ferguson recommended even if they don't get off to the best of starts, going short at 1/6 that he remains in charge come the end of upcoming season."
Moyes first comes up against his former club Everton at Old Trafford on December 3 before a visit to Goodison Park in April, with Sky Bet going 7/4 that United win both encounters.
United are 2/1 to finish up as runners-up and 6/5 to come third or below, while Moyes is 9/4 to win the LMA Manager of the Year award for 2013/-14.