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Pep Guardiola, who recently took his team to Abu Dhabi for a mid-season training trip, had met the billionaire owner.
The Spanish manager said that they had a long chat regarding their families, the club, etc.
Guardiola said: "I met Sheikh Mansour for the first time and I was really impressed that he knew absolutely everything about us.
"It was nice to meet him and spend a few hours together. We spoke about private lives, families, the club, about my impression after six months here. The owners are able to help us, not just me but the people at the club, the staff.
"They were so kind. He would like me to stay here for a long, long, long, long time. But of course, it depends on results.
"If we do not get good results they'd say, 'What happened? What can we do? What we need to improve?' That's the process and they are going to help us."
Prior to Guardiola, Manuel Pellegrini was in charge of the club. The Chilean manager was given a target of winning 5 trophies in 5 years.
While Mancini won 3 titles (2 EFL Cup, 1 EPL), the owners were reportedly not happy with the achievements.
On Pep's target, he said: "We didn't speak about target. They will be happy if we win. I know we have to win, but I have handled that in the past and I will do so in the present.
"All the managers and owners know that we are here because of the results. If the results are not good we are going to be replaced by another.
"It is important we are together, both departments, the chairmen, managers, and owners, that we have the same objective, the same target."