Royal Bank has acknowledged receipt of Midshires' letter and told the building society it is considering its request.
Under the terms of the agreement it signed with Royal Bank in the summer, Midshires is forbidden from holding formal talks or exchanging information with third parties.
Its board is also obliged to put Royal Bank's proposals to its members, although it does not have to recommend the deal.
At the time of the Halifax bid, Royal Bank said the agreement between itself and Midshires was legally binding. It also said it would not, any circumstances, increase its offer for Midshires.
A Midshires spokesperson said the building society had "no intention of breaking its agreement with Royal Bank". Instead, Midshires is relying on "common sense to prevail" at Royal Bank, according to a source close to the discussions.
If Halifax is the eventual winner in the struggle for the building society, Midshires' members could stand to net an additional windfall of pounds 150 each.