Kleyn will join his former defence coach Jacques Nienaber, who he worked closely with during his time at the Cape Town franchise, with Nienaber having joined the Irish province earlier this summer.
The 22-year-old has been signed on a three-year contract and will arrive in Ireland in August, subject to being granted a work permit.
The 6ft 8in second row has made 17 appearances for the Stormers and represented Western Province on 22 occasions.
Since 2012, the South African native steadily progressed through the Western Province ranks establishing himself as a key player, before going on to secure regular game-time with Super Rugby side, the Stormers, from 2014.
Recently appointed Munster director of rugby, Rassie Erasmus, said: "Jean is a hugely exciting prospect and a great addition to the province.
"I've seen his development first hand and with Jacques' extensive knowledge of him we are all up to speed on how Jean can contribute to the squad.
"A naturally athletic player we believe he will add further value to the pack and we look forward to welcoming him to Munster."
Kleyn is the Thomond Park side's fourth signing of the summer after Sam Arnold, John Andress and Darren O'Shea.
Munster begin their new Guinness PRO12 campaign with a trip to Scarlets on September 3.
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