City won their first league title for 44 years in dramatic style, stealing the crown off United with virtually the last kick of the campaign.
However, Mancini has been frustrated by City's lack of transfer activity, and believes that they may now be as low as fourth favourites.
"United start as favourites this year. We are maybe in second, third or fourth position," said the 47-year-old Italian.
"When you win something it is important to improve, but you should not buy new players 10 days from the end of the window. It should be two months.
"The championship finished on May 13. You needed to buy players before now. Now it is difficult because there are some teams who can't sell their players because they don't have time to buy anyone else. The next 20 days is very difficult in the market."
City face Chelsea in the Community Shield final tomorrow before beginning their title defence against Southampton a week later.