Liverpool lifted their first league title in 30 years last season while City won back-to-back titles over the previous two campaigns and secured a domestic treble in 2018-19.
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"I think that's natural for anyone who plays for the club or anyone who's from Manchester," Rashford told PA News Agency. "We have to use all that as motivation to keep improving as a team.
"But we can't lose sight of what's important, and that's our own team's development and to keep improving.
"You can find yourself sometimes looking too much elsewhere, but we play our best and we improve most when we concentrate on ourselves as individuals and as a team.
"The manager always wants us to do that and that's the best way to do it."
United failed to win a trophy for the third consecutive season, but secured a place in the Champions League with a top-three finish after a 14-match unbeaten run in the Premier League.
Rashford finished as the club's second-highest scorer last season with 22 goals but said he wants himself and the team to perform on a regular basis.
"It's obviously a big step," he added. "The lads worked hard and for me, it's what we deserved as a team. We've had some tough times this season and sometimes where we've been unstoppable.
"But we need to find that consistency, that's the biggest thing for us. We have quite a young team, so if we can find that consistency, that will be important for us.
"I missed four or five months and in my mind, I could have done much more this season... but for whatever reason, it didn't happen.
"I never lose touch with reality and I know next season, I have to come back better than I was last season."