Executives from the pounds 5.9bn bank are understood to have already held internal discussions about a bid - just days after Midshires was released from a pounds 630m offer from Royal Bank of Scotland.
Sources said the bank was likely to approach Birmingham Midshires next week with a proposal for talks that could lead to a bidding war with the Halifax and other interested parties.
Bank of Ireland is looking to expand its interests in the UK after successfully buying Bristol & West Building Society for pounds 600m last July. It has explicitly stated its interest in further expansion into the UK mortgage market.
Pat Molloy, the bank's chief executive, confirmed at the bank's last results that a UK acquisition was on the cards. Asked whether he had any plans in the UK market, he replied: "While we have nothing specific in mind we would certainly be interested in expanding in the area of mortgages."
Ironically, the Irish bank has significant ties with Royal Bank of Scotland, which demanded a pounds 5m payment after Midshires reneged on a deal struck last August to buy the society for pounds 630m. After discussing the deal with George Mathewson, chief executive of RBS, Midshires agreed to pay a further pounds 10m to RBS if the society was bought by the Halifax - but not if another bidder was successful.
Royal Bank of Scotland holds the majority stake in Citizens Financial Group, a bank based in New England on the eastern seaboard of the US. Bank of Ireland has a 23 per cent stake in Citizens. Both Dr Mathewson and Mr Molloy sit on the board of Citizens.
Halifax is expected to meet Midshires executives next week to discuss terms.