Owner Sheikh Mansour has spent upwards of £1 billion on his City project, which saw City claim their first league title for 44 years under Mancini.
However, following failure in Europe's elite competition for a second consecutive season, recent reports suggested the Italian's job was under threat with Jose Mourinho and Pep Guardiola been lined up for next season.
But, Mancini has revealed that although the club owners have been disappointed by the recent failure to reach the last 16, he will be given time to deliver a successful season.
Mancini said: "He (Al Mubarak) is happy with the way we are going in the Premier League, and we have the FA cup to come.
"Obviously they (owners) are so disappointed about the Champions League, and this is normal because our target when we started the competition was to get to the knockout stage - and anything can happen when you arrive there.
"But we didn't do well and for this reason I think they are disappointed but they understand it takes time to succeed in the Champions League."