The men spent over an hour dragging the 368lb (167kg) beast on board the boat off the coast of Cornwall on Sunday.
One of the men, Nick Lane, said they had submitted a record application after the Shark Trust calculated the weight based on the shark's dimensions.
"I was jumping up and down like a little school girl", Mr Lane said. "All hell broke loose."
Thresher Shark latest
"Suddenly the rods took off, they were absolutely screaming," he said.
Mr Lane, a garage owner from Washford in Somerset, said when they got the shark on board they measured it and took photographs before letting it back into the water.
John Richardson, Shark Trust conservation officer, praised the anglers for putting the shark back into the sea.
He said: "Here in the Northeast Atlantic Threshers are listed as Near Threatened on the IUCN Red List of Threatened Species.
"The Shark Trust applauds the anglers for returning the shark alive to the water, especially as it may well have been a potential record."
Mr Richardson said normally anglers would need to bring a shark to shore to be formally verified for a record, although sometimes informal records can be recognised.
To be the largest shark caught in British waters it will need to break the previous record set in 1982 by a 323lb (147kg) thresher shark.
Nick Simmonds, secretary of the British Record Fish Committee (BRFC), said he had been unable to verify if a claim had been formally made yet.
Mr Richardson said he did have a record of their catch but the Shark Trust did not confirm or verify catch records.
A group of amateur fishermen have landed what they believe to be a record-breaking thresher shark.