Ulster Bank has begun the process of writing to its 900,000 current and deposit account customers to give them six months' notice to move their accounts to another institution where required or close them.
The letters are being sent on a phased and rolling basis in order to try to ensure that the process is orderly.
Ulster Bank said last year that it was going to wind down its operations in the Republic of Ireland.
It is selling €7.6 billion of assets including performing non-tracker mortgages, performing SME loans, its asset finance business and 25 branches to Permanent TSB.
AIB is also buying €4.2 billion in performing corporate and commercial loans from it.
However, none of the deals include the 900,000 current and deposit accounts which Ulster Bank currently has.
Instead, those account holders, including 360,000 primary active personal accounts and 300,000 deposit accounts, will either have to move them to a new institution or close them.
Ulster Bank said that customers do not have to take any action until they receive a letter or email from it.
It also said that its branches are to remain open during the process to assist customers who need in-branch support.
Over recent weeks, concern has been expressed by regulators and unions that the banking system is not fully prepared for the mass switching process by Ulster Bank and KBC Bank Ireland customers.
Yesterday, KBC said it had begun contacting customers on a phased basis to ask them to close their current accounts within 90 days.
In particular there are worries within the sector about the need for customers to tell utility providers, employers and others who use direct debits, standing orders and make salary payments into accounts of their new banking details.
But Ulster Bank said it is taking a responsible and prudent approach that will include regular monitoring and evaluation of the progress throughout the process.
It also said it recognises that for customers, particularly long-standing ones, choosing and moving to a new provider will require effort, time and support.
In a booklet that will be issued to customers, the bank has warned of the consequences of not closing accounts before the date notified in the letter.
"You will no longer be able to use your account," it states.
"Any Direct Debits, Standing Orders or future dated payments will not be made. Any payments into your account such as your salary, pension, child benefit, social welfare will not be paid into the account. They will be returned," it added.
The booklet says that customers who hold any other products like loans, mortgages or credit cards will be written to separately.
Last month, the bank said it would close counters in all of its branches at 1pm from the start of July, as it prepares for its withdrawal from the Irish market.
It said this will allow staff to concentrate on helping customers face-to-face, who may need support ahead of the closure.
From 1 July, the bank will also stop counter foreign exchange services, meaning it will no longer offer foreign exchange cash and FX drafts - but it said customers can still lodge foreign cheques.
It is also to close to new mortgage applications from existing customers from 10 June, and it will no longer offer approval in principal for existing customers from 29 April.