British fishermen believe UK waters are overfished by other EU countries and the EU’s Common Fisheries Policy (CFP) means they are also restricted as to how much they can fish.
The strict EU regulations mean that some British fishermen are forced to throw fish back into the sea.
Lib Dem MP Alistair Carmichael asked the Prime Minister: “Members of the Scottish Fishermen’s Federation and the National Federation of Fishermen’s Organisations are jointly lobbying Parliament today.
“They ask for a very simple undertaking from the Prime Minister.
“Namely, that in negotiating the Brexit deal she will not trade away the right to fish in UK waters. Will the Prime Minister give our fishermen that undertaking?”
In response, Mrs May said: “I absolutely recognise the importance of the fishing industry which it play across the UK but particularly in Scotland.
“Can I reassure him? As we are going through these negotiations we have been very clear that once the EU rules no longer apply to the United Kingdom, we will be an independent coastal state.
“And we will be making those decisions. We will be controlling access to our own waters and we will be seeking to gain a fairer share of quotas.”
But Tory MP Ross Thomson said First Minister Nicola Sturgeon’s bid to maintain close ties with the European Union after Brexit will “betray” Scottish coastal communities.
Mr Thomson told the audience at a Fishing for Leave event the decision by the SNP to back a second Brexit vote was an attempt to keep Scotland under EU rule.
The Tory MP said: “I am astonished that SNP members representing fishing communities would seek to betray fishermen, to drag them back into the Common Fisheries Policy and disregard their views altogether.
“It is vital we keep speaking up for Brexit and for what it will do to revitalise our fishing industry and our coastal communities.”
Mrs May has also hit out at the Scottish First Minister for her efforts to keep the country tied to the EU customs union.
The Prime Minister warned keeping Scottish trawlermen in the CFP would be a betrayal of Scotland.
Delivering her speech at the Conservative Party Conference in Birmingham, Mrs May said: “Our proposal would be good for our rural communities, getting us out of the Common Agricultural Policy. It would be good for our coastal communities.
“We would be out of the CFP, an independent coastal state once again. And with the UK’s biggest fishing fleets based in Scotland, let me say this to Nicola Sturgeon.
“You claim to stand up for Scotland, but you want to lock Scottish fishermen into the CFP forever. That’s not ‘Stronger for Scotland’, it’s a betrayal of Scotland.”