The Italian has added attacking midfielders Eden Hazard, Marko Marin and Oscar to his squad so far this summer for a combined fee of around £65 million.
Chelsea will face City in the Community Shield on Sunday and Di Matteo predicted that they would prove the team to beat as defending champions, with United not trailing far behind.
"[Manchester City] are last season's champions," Di Matteo said. "They'll be the team to beat, even though United were very close to them.
"Those two teams are probably a little bit ahead, but the champions we want to chase."
In addition to this weekend's pre-season curtain-raiser, Chelsea will compete in the Uefa Super Cup and Fifa Club World Cup this season as a result of their FA Cup and Champions League victories.
Di Matteo recognised the opportunity to challenge for extra silverware but also that his team must improve their final Premier League position after finishing sixth during the 2011-12 campaign.
"Our biggest aim, or our ambition, is to be able to be challenging at the right end of the season for the trophies that are still available," he continued.
"We are going to be in a couple of competitions which are new to us, so they're an opportunity for the club and the players, but we certainly have to do better in the Premier League."
Chelsea adopted a largely defensive style as they secured European glory last season, but owner Roman Abramovich is said to admire the 'tiki-taka' philosophy that has helped Barcelona to dominate domestic and continental competition in recent years.
Di Matteo was certain that his side, aided by their new signings, would entertain Blues fans but warned that no team could truly replicate the football played by the Catalan club.
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"Everybody would like to play like Barcelona, but there isn't another team in the world who play like them," he commented.
"It's a culture, a way of life, a style they've produced through the club for many, many years. And there's only one Lionel Messi around.
"The owner's will to win has never decreased. He wants to win, and he backs it up.
"It will be a challenge for us, for sure, to match the best in Europe and I'm not preaching we'll be this or that. We're going to be trying to win in the way we're capable of. We'll have some good games, I'm sure.
"I'm excited by these new players. You look at what they've done at previous clubs, and trying to bring them together to express their qualities for the team will be exciting.
"But I don't think you can expect a completely different Chelsea overnight."