A merger between the two companies would create Europe's 12th biggest bank, with a market value of about £15 billion ($21 billion).
In a statement to the London stock market, Bank of Ireland said talks were at an early stage. "No formal discussions have yet taken place. Accordingly, there can be no certainty that a transaction will result," it said.
Bank of Ireland is proposing to create one merged company, with its head office in Dublin and its main stock market listing in London. The Bank's confirmation of its bid approach came after press reports over the weekend.
Abbey has been seen as a likely candidate for a merger or takeover for the last two years. A planned tie-up with Bank of Scotland fell through last year, and a hostile takeover attempt by Lloyds TSB was blocked on competition grounds. This summer, Abbey was reported to be holding merger talks with National Australia Bank, which already owns the Clydesdale, Yorkshire, and Northern banks in the UK.