Some of the divers came so close to the female shark in the waters off Hawaii that they were able to touch it.
The shark, nearly 20ft (6m) long and weighing an estimated 2.5 tonnes, is believed to be one tagged by researchers about 20 years ago called Deep Blue.
It was attracted to the area off Oahu’s south shore by the carcass of a dead sperm whale.
One of the divers, Ocean Ramsey, told the Honolulu Star Advertiser that they had been filming tiger sharks feeding on the whale when the shark arrived.
“We saw a few tigers and then she came up and all the other sharks split, and she started brushing up against the boat,” she said.
“She was just this big beautiful gentle giant wanting to use our boat as a scratching post. We went out at sunrise, and she stayed with us pretty much throughout the day.”
Ms Ramsey said the shark was “shockingly wide” and could be pregnant. Deep Blue is believed to be about 50 years old and has its own Twitter account.
Great whites are rarely seen around Hawaii as they prefer cooler seas.
Ms Ramsey said older, pregnant great white sharks were the safest to swim near but cautioned against swimming anywhere where sharks were feeding.
She said sharks would only attack humans if they were curious or if they mistook people for their normal prey, the Honolulu Star Advertiser reported.