The England international, who joined City from Aston Villa for £26million in 2010, is in the final year of his current deal.
The 28-year-old has played a prominent role for the Barclays Premier League champions in the first half of the season, featuring in all but two of their 24 games in all competitions.
Milner's future had been the subject of speculation due to his lack of starts last season but his prominence this term - starting 13 times - appears to have changed that.
The development comes after goalkeeper Joe Hart admitted this week he was also ready to sign fresh terms.
Pellegrini said: "Milner, I hope that he will sign also
We will see, same as Joe, in the next days.
"I don't know how many days before we can have this as official news but I suppose that also he will sign and continue here because he also wants to stay."
Pellegrini first revealed that a deal with Hart was close a week ago but he could not confirm when anything would be finalised.
The Chilean, speaking at a press conference to preview this weekend's game at Leicester, said: "It is very close, I suppose.
"As you know, Joe wants to stay here
We also want him to be our goalkeeper so I hope it will be ready as soon as possible
I don't think that we are going to have any problems."
Pellegrini also had no news on a possible new deal for Frank Lampard, whose loan deal from sister club New York City expires at the end of the month.
The former England midfielder has proved a success in his short stint at City, leading to speculation his stay could be prolonged to cover Yaya Toure's involvement in the African Nations Cup.
But, once again, Pellegrini would say nothing beyond the fact the matter would be looked at sometime in December.
He said: "I said during December
For the moment, no (news)."
Source : PA
Source: PA