What appeared to be an adult and an infant, due to the size of their fins, was filmed swimming in the shallow waters of Keem Bay on Achill Island in County Mayo.
Although the species is harmless, onlookers feared it might have been dangerous with a child heard screaming “oh my god”.
Alan O’Neill, meanwhile, wrote that it was “like a scene from Jaws” when sharing the clip on Twitter.
Basking sharks – which often reach 26 feet in adulthood – are just behind the whale shark in terms of size, but feed on microscopic plankton.
It is a cosmopolitan migratory species, present in all the temperate oceans of the world.
The name comes from its habit of feeding by filtration on the surface of the water, giving it the appearance of a lizard.
The bay was previously home to the world’s largest basking shark fishery, with up to 50 sharks a day often caught before the trade ended more than 30 years ago.
In the fishing industry, its flesh and fins are used for food, including soup, and its skin for leather and liver for oil.
In recent years, however, there have been several sightings off the coast of the United Kingdom and the Republic of Ireland, including when one person has been feeding at the same spot in Keem Bay every day for a month.
In April, a 16-foot basking shark was briefly seen swimming alongside a boat in Portloe Bay, Truro, Cornwall.
That same month, a paddleboarder was captured surrounded by 20-foot basking sharks in spectacular photos taken in Porthcurno, Cornwall.
The video shows the sharks circling the paddleboarder with photographer Michael Amos, 20, saying that far from being scared, it was a “privilege” for him to get a view of the incredible marine predators.
The marine and natural history photography student said: “I spent over an hour photographing them in perfect conditions.
“The clear water of Porthcurno allowed me to have a breathtaking view of them swimming on the surface. “
The species does not hibernate and is active year round, increasing the likelihood of being seen, especially during the warmer months, as in winter it typically dives to depths of up to 3,000 feet.
However, it is listed as vulnerable by the IUCN (International Union for the Conservation of Nature) list.
Besides direct catches, bycatch in trawls has been one of the many threats to basking sharks.