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.
‘The pure simplicity of the way Slot has managed the squad is probably the biggest thing I could say about him. It’s not broken, so let’s get on with it’: Liverpool legend full of admiration for Jurgen Klopp's successor at Anfield
‘I trained at Spurs and thought they’d give me a chance. But I received a letter thanking me but saying they didn’t think I was good enough – I was gutted’: How Tottenham missed out on signing Wales legend John Toshack
