The 29-year-old missed Liverpool's 2-1 Premier League defeat by Crystal Palace on Sunday, their first loss under manager Juergen Klopp.
"It's always disappointing to miss any games - and obviously missing last weekend and the England games is disappointing," Milner told the club's website.
"It shouldn't be too long and hopefully I'll be back for the next game (against City).
"You never like to put a time frame on it because it's a hamstring and they're always difficult and a bit unpredictable, but hopefully I'll be alright," he added.
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Liverpool sit 10th in the table after 12 games, nine points adrift of league leaders Manchester City.
(Reporting by Shravanth Vijayakumar in Bengaluru. Editing by Patrick Johnston)
