More than 900 workers at Quinn Insurance have applied for voluntary redundancy.
The company was put into administration by the High Court in Dublin in March.
Quinn Insurance is shedding 900 staff at locations on both sides of the border including Navan, Cavan and Dublin.
Two hundred of the jobs will go in Northern Ireland at Enniskillen and Derrylin.
It will be several weeks before the applicants will be told whether or not they will qualify.
In March, Dublin's High Court heard the company had significantly breached its solvency ratios.
The company had moved from a position where it had an excess of assets over liabilities of more than 200 million euros to a position where it had 200 million euros of liabilities over assets.
At the end of April the company's administrators announced the need to cut the posts.
Between all the centres affected, more staff have applied for voluntary redundancy than jobs are being lost, but the administrators said it is too early to say whether this will cancel out the need for any compulsory redundancies.
The company now has to assess the grades and locations of those who have applied.
Under the terms of the redundancy programme, a member of staff who has been with Quinn Insurance for five years would receive 31 weeks pay.
In order to move Quinn Insurance into profitability, the administrators have said the job losses are necessary to cut almost £30m off the annual wages bill.
The job cuts will be carried out over a 15-month period up until the middle of 2011.