Sturridge missed England's 2-0 victory over Switzerland on Monday after picking up the problem in a training session following last week's friendly win over Norway.
Rodgers, who earlier criticised the handling of his striker, revealed his "disappointment" at losing the 25-year-old for the early part of the campaign at a media conference on Thursday.
Should Sturridge be sidelined for three weeks, he would miss six games - including Liverpool's opening UEFA Champions League fixture with Ludogorets on Tuesday and the first Merseyside derby of the season against Everton on September 27.
Sturridge is not Rodgers' only injury concern ahead of his side's clash with Aston Villa on Saturday, with Joe Allen doubtful and Glen Johnson, Martin Skrtel, Jon Flanagan and Emre Can all sidelined.
However, Adam Lallana is in line for his Liverpool debut after recovering from a knee problem picked up in pre-season.
"Emre Can could be out for up to six weeks with an ankle injury," Rodgers added.
"Glen Johnson's still got a week to go, [and] Jon Flanagan is edging towards [fitness] but not in the squad yet.
"Adam Lallana is back in, [but] Joe Allen has an issue with his knee, we'll assess him."
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