The 64-year-old insists that the club's fans will be behind the Chilean manager, who was finally appointed on Friday following last season's dismissal of Roberto Mancini
The 64-year-old became Roberto Mancini's assistant manager in December 2009 and took charge of City's final two Premier League games last season following the Italian's dismissal.
In the wake of Pellegrini's appointment, Ruben Cousillas was brought to City as the Chilean's assistant, while the club have named Xabier Mancisidor as their new goalkeeping coach, and Jose Cabello as fitness coach.
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"It's a compliment and a great honour that I'm stopping with the club. I'm very pleased that Mr Pellegrini wants me to stay.
"Obviously, he'll be setting his own standards and his own philosophy. He'll have his own principles about how to play which is exciting. He needs to speak to a few players, which he's going to do, and then he'll get a better picture.
"I know the Manchester City fans and they'll be behind the new manager and they'll get behind the players. It's been a difficult couple of weeks with what's happened. We've got to consolidate now, look to the new season and get on with it."
Pellegrini insists that Kidd has a major role to play at the club and is looking forward to City's pre-season campaign.
"It is important to me that Brian remains part of the coaching team," the new boss added. "He has great football knowledge, knows the league, the players and will be a big part of what we try to achieve. We have spoken and we are all excited to work together."