The 38 year-old joins the Foxes on a one-year deal.
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Begovic spent last season with Everton, where he acted as cover for Jordan Pickford, having spent the previous year as QPR's senior No1.
The veteran posted to X after putting pen to paper: "Absolutely delighted to have joined @LCFC.
"A massive club full of tradition and history and incredible fanbase. Looking forward to the challenge ahead. Let’s go!"