The 26-year-old - signed in the close-season from Charlton Athletic - picked up the injury in training on Friday prior to Leicester's 2-0 defeat to Swansea City at the Liberty Stadium.
Leicester boss Nigel Pearson told the club's official website: "He had a nasty break of his finger on Friday which needed an operation. He'll be out for a few weeks.
"It was pretty straightforward in terms of the surgery and he'll probably have another operation again this week and then we'll see how he is.
"The first prognosis would be that it will be about three or four weeks."
Hamer made his Premier League debut for Leicester against Stoke City last month - his clean sheet helping to secure a 1-0 win at the Britannia Stadium.
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