Mourinho and Guardiola clashed while in charge of Real Madrid and Barcelona in La Liga respectively from 2010 to 2012.
"For two years I was with Pep in a league where the champion was either him or me," said 53-year-old Mourinho.
"If I were to focus on him and he were to focus on me, someone else would be champion."
During their time together in Spain, Guardiola won the Champions League and the Fifa Club World Cup. He and Mourinho each won one La Liga title, one Spanish Cup and one Spanish Super Cup.
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Portuguese Mourinho was appointed United manager last week following Louis van Gaal's sacking, while Guardiola takes over from Manuel Pellegrini at City.
Mourinho and Guardiola's meetings in Spain became increasingly antagonistic, and Mourinho poked Guardiola's assistant Tito Vilanova in the eye during a melee in a Spanish Super Cup tie in 2011.
Guardiola's first game as City manager will be against United in a pre-season friendly in Beijing on 25 July.
Mourinho visited United's training ground at Carrington for the first time on Monday, when he met club legend Sir Bobby Charlton.
Mourinho has not spoken with assistant manager Ryan Giggs about his future at Old Trafford, with long-time assistant Rui Faria set to be appointed.
It is understood Mourinho wants 34-year-old midfielder Michael Carrick, who is out of contract this summer, to stay for another year, although a deal has not been been signed.
New Manchester United manager Jose Mourinho says it would be a mistake for him to focus on Manchester City boss Pep Guardiola next season.