The champions face Liverpool, Chelsea and Manchester City in three of their first five games under Moyes, and Red Devils legend Neville has said that they must be wary about slipping up early on.
"You wouldn't want to fall too far behind a Jose Mourinho team because they would have the nerve inside the club to withstand the pressure," The Mirror quotes Neville as saying.
"Where we lost out to Mourinho's [Chelsea] teams at United in those early years was because we left ourselves too much to do. When we did get into our stride in January, February and March, the problem in the Mourinho years was we were too far behind to mount a challenge.
"So the critical thing this year will be to make sure that, at the turn of the year, you're in contention. Make sure you don't fall back too far behind."
Moyes begins life as United boss in the Premier League against Swansea City away from home on August 17.