Senior Brussels diplomats were quoted in The Telegraph saying EU fishing fleets must be granted access in exchange for a UK-wide customs union forming part of the backstop deal.
They reportedly want to “extract a high price” for the concession if a trade deal is not reached before the expected transition period ends in December 2020.
The claims have raised anger from Brexit supporters.
Sammy Wilson, Brexit spokesman for the minority government’s partner the Democratic Unionist Party, said Mrs May must face down the demand and keep her promises to the electorate.
He added allowing EU fishing vessels access to UK waters was “going to raise the ire of Scottish MPs”.
“She just doesn’t have a battle with the DUP, she has a battle with a substantial number of her own party,” Mr Wilson told Sky News.
Former UKIP leader Nigel Farage also claimed that “fishing is the acid test of Brexit”.
“We must take back control of our waters from Brussels,” he announced.
Niels Wichmann, chief executive of the Danish fishermen’s association, told The Guardian last week that “the issue of access to waters is part of the trade negotiations”.
“That is what we have said from the start,” he said.
“We discussed this with [the EU’s chief Brexit negotiator] Michel Barnier when he came to Denmark. And he confirmed that this was also his position.”
Environment Secretary Michael Gove promised last month that the UK would “take back control of our waters” after Brexit.
“The Common Fisheries Policy has damaged the UK’s fishing industry and our precious fish stocks,” he said.
“The Fisheries Bill will deliver a sustainable fishing industry, with healthy seas and a fair deal for UK fishermen.”