It would be the first breach in the male-dominated institution. The Bank employs no women at the top, as executives or non-executive directors.
The appointment to the Court, the Bank's board, would be made shortly after the announcement - expected in mid-January - of the new Governor, who will take over from Robin Leigh- Pemberton in the summer. A final shortlist for the governorship has been drawn up and it is thought that Downing Street has approached the favoured candidate. If the candidate accepts, the name goes to the Queen for approval.
Eddie George, the present deputy governor, is favourite, but Sir David Scholey, chairman of Warburg, and Sir Christopher Hogg, chairman of Reuters, are other possibilities, along with Sir Dennis Weatherstone, the head of JP Morgan, though he has switched from British to US citizenship.
The Court of Directors the new governor will inherit is made up from industrialists, City figures, career Bank of England people and one trade unionist.
Women of the same seniority in industry and the City are still thin on the ground. But names expected to be considered include Lady Howe, wife of the former Tory cabinet minister, and Rosalind Gilmore, who heads the Building Societies Commission - though she may not be eligible as she holds a senior Whitehall post.
Other possibilities include Gail Redwood, company secretary of British Airways and wife of John Redwood, the local government minister, Kathleen O' Donovan, finance director of BTR, and Denise Kingsmill, the lawyer.
The Prime Minister has been insisting that one woman is included on shortlists submitted for all senior appointments under his influence. The Bank has been pinpointed as an institution where a woman directorship should be given a high priority.
The Prime Minister came under strong criticism early in his period of office for not moving women into the cabinet fast enough, and a woman director at the Bank of England would signal an effort to break down barriers in Britain's most important financial institution.
The appointments of three directors, Sir Brian Corby, the Prudential chairman, Lord Haslam of Bolton and Sir David Walker, expire or come up for renewal at the end of February.