Police and other rescue workers had hosed down the surviving whales at Noordhoek Beach to try to keep them alive.
The South African Press Association quoted Craig Lambinon, a spokesman for the National Sea Rescue Institute, as saying authorities had considered whether to try and refloat the whales, which washed up on the beach on Sunday morning.
"Seven are in poor health," SAPA quoted Lambinon as saying. "We are still trying our best to save them, but those that can't be saved will be humanely euthanized."
One whale was being transported to a naval base and will be taken out to sea, according to authorities.
"At this stage the first whale is on its way on a trailer to the naval base," Lambinon said. "There are a remaining five whales in good health and we are going to attempt to do the same for them."
He appealed to the public to stay away from the beach because enough workers are there, trying to help the whales.
In 2009, authorities in the Cape Town area removed the carcasses of 55 whales that beached themselves and had to be shot despite the frantic rescue efforts of hundreds of volunteers.