Mancini's team face Chelsea in the Community Shield on Sunday and begin the defence of their title the following weekend against Southampton.
However, the Italian believes that the squad still have a lot of work to do before they reach their targeted fitness levels despite a busy pre-season schedule.
Mancini told the club's official website: "The games against Oldham and Limerick were very important, but I think we need another three or four weeks for these players to get back to full fitness.
"We will all need to work very hard at Carrington. We will start to work together, the whole team together as we prepare for the season ahead."