According to a cricket website, the 32-year-old talented batsman is expected to join the side soon as clearances for his stint with the team have been obtained.
Amla, who remained unsold at the auction in February, is yet to make his debut in the lucrative cash-rich league. Earlier this year, he had stepped down as South Africa's Test captain in the middle of the England series.
He has scored 2446 runs at an average of 31.35 in his 88 T20s.
Marsh picked up the injury during his side's six-wicket defeat against Kolkata Knight Riders on April 19. He experienced some discomfort while batting and fielding during Kings XI's win against Gujarat Lions on Sunday.
Image: South Africa's Hashim Amla
Photograph: Anindito Mukherjee/Reuters
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