A spokeswoman for the UK Post Office, which handles passport applications, said: "We have seen an unusually high number of people in Northern Ireland seeking Irish passport applications, though we do not have exact numbers or a breakdown by branch."
Britain voted in a referendum last Thursday by 52 per cent to 48 per cent to leave the 28-nation EU, whose citizens enjoy free movement rights.
They can also seek employment in other EU states without the need for a work permit, use public healthcare facilities across the bloc and benefit from welfare and other rights.
Discussion about applying for an Irish passport has been rife on social media.
Twitter user @Saraita101 posted a picture of a sign put up at the central Post Office in Belfast.
"Due to the increased demand for Irish Passport Forms we have currently run out. We have an emergency order to meet the extra demand coming on Tuesday 28th June," it read.
Geographically most applications have been made in the British province of Northern Ireland. Anyone born on the island of Ireland before January 1, 2005 is entitled to Irish citizenship.
There were 430,000 Irish-born people in Britain while one in four people in Britain have Irish heritage, according to 2011 Census figures cited by the Irish Department of Foreign Affairs and Trade.
The ministry played down the urgency of the situation, and said it should not slow down the processing of passport renewal applications for Irish citizens.
"There is no urgent need for UK passport holders to look into applying for a possible other EU member state passport," it said.
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