Britons trying to hang on to EU citizenship have inundated Ireland's embassy in London and post offices in British-run Northern Ireland with passport enquiries and requests for application forms, the Irish foreign office said on Monday.

Post offices ran out of forms and the embassy fielded more than 4,000 passport enquiries compared to the 200 a day it usually gets, a diplomatic source told Reuters.

Anybody born in the Irish Republic or Northern Ireland, or with an Irish parent or grandparent, is entitled to an Irish passport — about 6 million people living in Britain.

"Following the UK referendum, there has been a spike in interest in Irish passports in Northern Ireland, Great Britain and elsewhere," Irish Foreign Minister Charlie Flanagan said.

"The increased interest clearly points to a sense of concern among some UK passport holders that the rights they enjoy as EU citizens are about to abruptly end," he said in a statement.

Flanagan warned that a surge in applications would place significant pressure on turnaround times at passport offices and could affect those with imminent travel plans.

Scrabble board A Scrabble board expresses what’s on the minds of many in the U.K. Photo: REUTERS/CLODAGH KILCOYNE

A member of Northern Ireland's largest Irish nationalist party, Sinn Fein, called on the government in Dublin to open a passport office in Belfast after post offices ran out of forms and were unable to meet demand until more arrived.

Even pro-British lawmaker Ian Paisley Jr., the son of the firebrand preacher-politician of the same name who for decades cried "No Surrender!" to Catholic nationalists' desire for closer ties with the Irish Republic, advised constituents to apply for a second passport.

Northern Irish citizens can hold both an Irish and British passport.

Ireland's embassy in Paris has also had a record number of requests, according to English-born Iain McKenney, who has been living in France for eight years and phoned the consulate on Friday to ask if he was eligible.

"The best news I got this morning was when they told me 'Welcome to Ireland.' I'll submit the application immediately," said McKenney, 41, who holds British and Canadian passports but wants to retain one for the European Union.

Elsewhere, views are mixed among the estimated 400,000 British expatriates entitled to dual citizenship if they have lived in France for five years or have a French spouse.

Anne Wilding, a 63-year-old British pensioner who has lived in the tiny western French village of La Petite Breille for 15 years, said she was born British and wanted to keep her nationality but may apply for French citizenship as well if Britain does not good get a good deal on leaving the EU.

Some Britons are applying for Belgian passports as longtime residents of Belgium or applying to the countries of their European spouses. Mayors of boroughs and towns around Brussels told Le Soir newspaper their town hall staff had been bombarded by requests on Friday from Britons seeking citizenship.

In London, where the population of 8.6 million is among the most cosmopolitan in the world, younger workers who mainly voted for Britain to remain in the EU were also mulling their options.

"I'm not happy with the result," said Miriam Sottile, a 25-year-old English teacher whose father is Italian and who is now planning to apply for an Italian passport.

"I feel like I'm European. I want to stay in the EU and I don't know what the limitations will be. I want to have the same opportunities I’ve always had."