Halifax's offer represents a premium of 24 to 29 per cent over RBS's offer of between pounds 605m and pounds 630m. David Gilchrist, Halifax's group secretary, said RBS's bid "significantly undervalued the business".
Royal Bank of Scotland had earlier said it would not raise its offer "under any circumstances".
City experts said the Halifax offer would be good news for more than 1 million Birmingham Midshires' members - who each stand to net an additional windfall payment of up to pounds 150 - but could be bad news for the building society's 2,000 employees.
Royal Bank of Scotland has said it would preserve the Birmingham Midshires brand as well as its 115-strong branch network. It has also guaranteed jobs for Birmingham Midshires' employees for three years.
City analysts believe Halifax is likely to cut jobs and close branches in order to justify the premium it has put on the building society.
Halifax has pledged to keep the Birmingham Midshires brand for at least three years, but has made no guarantees about jobs or the branch network. Mr Gilchrist said no such guarantees would be forthcoming, although he stressed that Halifax's previous large acquisitions - Clerical Medical and Leeds Permanent Building Society - were completed without the need for any compulsory redundancies.
A Birmingham Midshires spokesperson said the directors were considering Halifax's offer. On the subject of possible job losses, he said: "It has always been integral to Birmingham Midshires' board to add value to the society and its key stakeholders. That includes our members and our 2,000 employees."
Halifax said it had been considering making an offer for Birmingham Midshires for "several weeks", but only approached the building society on Friday, when Jon Foulds, Halifax's chairman, held informal talks with John Leighfield, his counterpart at Midshires.
Under the terms of its agreement with Royal Bank of Scotland, signed last year, Birmingham Midshires cannot hold formal talks with a rival bidder. It is also obliged to put Royal Bank of Scotland's offer to its members, although it does not have to recommend the bid.
Members of Birmingham Midshires are due to receive details of the Royal Bank of Scotland offer in April.
They were scheduled to vote on the proposed deal at a special meeting in June and the takeover was expected to be completed in the early autumn.
Halifax said that it intended to match Royal Bank's existing timetable as far as possible.
Under the terms of the offer from Royal Bank of Scotland, Birmingham Midshires' 1.1 million savers and 130,000 borrowers each stood to collect windfalls worth up to pounds 600.
The Save Our Building Societies campaign, which has opposed the Royal Bank of Scotland offer for undervaluing Birmingham Midshires, described Halifax as "nothing better than an enormous carpetbagger".
Outlook, page 19Reuse content