The 32-year-old defender is believed to be close to signing for City in a deal worth £4.2 million.
Pellegrini is in the market for a centre-back after seeing his captain Vincent Kompany injure his groin in their victory over Newcastle United.
"I expect progress," the Chilean said. "I hope that we can get one more defender for the team.
"Maybe he’s the name we want. We will see what will happen with him."
The 59-year-old wants the England man to stay and fight for his place at the Etihad Stadium, and hopes he will provide competition to the likes of Yaya Toure and Fernandinho, who have started all three of City's league games so far this season.
He explained: "All the players have a future here. At the moment we have all of the squad fit.
"We have 23 players it’s impossible for all of them to be playing.
"Gareth Barry is an important player but we also have Toure, Fernandinho, (Jack) Rodwell and Javi Garcia, so we have four players in that position."
