The windfalls, which will be paid on or before 25 August, follow the Bank of Ireland's pounds 600m takeover of the building society.
Savers with less than two years standing at Bristol & West, as well as borrowers, will receive preference shares rather than cash, while savers who are under 18 years old will receive a statutory cash bonus.
Members will be sent voting forms on the merger, and the transfer document which gives full details of the terms of the deal, by 14 March.
Mr Burke, who earns a base salary of pounds 200,000, was paid a bonus of pounds 80,000 last year. The building society announced pre-tax profits of pounds 86.6m, an increase of 10.3 per cent compared with 1995.
Bristol & West has 1.1 million members and around 430,000 of them have had accounts for more than two years and as a result are due to be paid cash.
The exact payment for individual members will depend on the balance in accounts on 14 April 1996, the day before the merger was announced, and 25 June, 1996. There will be a fixed cash payment of pounds 500 each with the remainder made up of 6.5 per cent of the savers' balance.
Those who had less than pounds 100 in their accounts on 14 April 1996, but who have held accounts at Bristol & West since 31 December 1994 will receive a flat cash distribution of pounds 250 each.