Pellegrini has been favourite to succeed Roberto Mancini since the Italian was released by City before the end of last season, but issues over unpaid wages and compensation have prevented City officially announcing the Chilean.
Malaga have now confirmed Bernd Schuster will succeed Pellegrini at the Spanish club and Mills insists the manager-in-waiting must be installed at the Etihad Stadium swiftly in order to accelerate preparations for next season.
"I think they have to get this done sooner rather than later," he told Sky Sports News. "This has been the worst kept secret in football. It has been known about, and talked about, for the past six weeks.
"There are obviously financial implications between the two clubs to be sorted out, which look like they are all done. But if you are going to look at who have been the best football club over the last 20 to 25 years - Manchester United - they have always got their business done very early on, before pre-season.
"If you are going to have your squad pretty much together as pre-season starts, you all start together and you start that team bonding and that methodology of how you are going to play and how you are going to coach.
"When you start bringing in players towards the end of the transfer window, even two or three games into the start of the season, it makes it very difficult to fit those players into what is almost a settled squad."
Buy, buy, buy
City have already recruited Spain winger Jesus Navas from Sevilla and Brazilian midfielder Fernandinho and Mills - who says transfer moves will be made by committee rather than individuals - expects City to splash the cash on several more big-name signings.
He said: "I would still expect two or three big names to come into Manchester City if they really want to have a go and wrestle that trophy back from Manchester United. But also Chelsea with Mourinho coming in; they are going to be right up there for favourites for the Premier League."
Mancini's reign was blighted by troubles within the squad and Mills says Pellegrini's priority will be to unify a squad, which has undergone huge changes in recent seasons.
"Pelligrini has a difficult job," he said. "There are lots of different factions within that Manchester City dressing room. He has got to pull all those players together. We have seen on several social media sites since Mancini left, it was not a happy camp.
"The first job for Pelligrini to do is to get all those players together, find a common language, a common denominator and get everybody onside."