It is understood that clearances for Amla's stint with the team have been obtained and ESPNcricinfo has learnt that he is expected to join the side soon.
Amla, who stepped down as South Africa's Test captain in January this year, is yet to make his IPL debut. The batsman was part of the player list for the 2016 auction in February, with a base price of INR 1 crore, but remained unsold.
Overall, Amla has played 88 T20s scoring 2446 runs at an average of 31.35 and a strike rate of 125.95. He scored two fifties as an opening batsman for South Africa in the recent World T20, tallying 120 runs in four matches in the tournament at a strike rate of 134.83.
Marsh reportedly picked up the injury in the game against Kolkata Knight Riders on April 19, and has been carrying a niggle since. He batted and fielded in Sunday's game, when Kings XI beat Gujarat Lions, with some discomfort.
Arun Venugopal is a correspondent at ESPNcricinfo. @scarletrun