8 August 2016 Last updated at 14:23 BST
The number of people in Northern Ireland applying for the documents rose by more than 60% in July, compared to the same period last year.
In the first full month since the UK's vote to leave the EU, there were 6,638 applications for Irish passports from people living in Northern Ireland.
DUP MP Ian Paisley Junior has said he is "relaxed" about the rise in the number of applications for Irish passports.