Aguero, who has scored 19 goals in 20 appearances in all competitions for City this season, sustained the problem in his side's 6-3 Premier League victory over Arsenal on Saturday.
Initially, it was hoped that the Argentina international would miss around a month, but Pellegrini has revealed this timeframe could now be doubled.
Should Aguero be out for two months, he would face a race against time to be fit for City's crunch last-16 UEFA Champions League tie against Barcelona.
When asked to provide an update after City's 3-1 League Cup win at Leicester City on Tuesday, Pellegrini replied: "I am not a doctor so it's very difficult.
"The doctor has said that he has at least one month or eight weeks (out). We'll see, I don't know how many weeks but it will be at least month."
In another injury blow to Pellegrini, Aguero's compatriot and team-mate Pablo Zabaleta limped off at the King Power Stadium after just 11 minutes.
The Chilean confirmed Zabaleta has a "problem with his hamstring", which will be assessed on Wednesday.
‘If I was Bournemouth, I’d be thinking if they can get past Manchester City, there’s no-one to stop them winning the FA Cup for the first time ever’ Former England star believes the Cherries have the ability to make history
‘I wasn’t concerned I’d done anything wrong – I stated a fact. It was taken out of proportion and people have recognised that they misread the situation a bit’ Gary Lineker gives FourFourTwo his thoughts on BBC suspension for asylum seeker tweet
