The final decision will be made by Mark Carney, King's successor, who takes the post July 1, The Guardian reported.
King received a wave of criticism earlier in his term when he decided remove prison reformer Elizabeth Fry from the country's banknotes by 2015.
Fry, the only other woman to appear on a banknote other than the queen, will be replaced by Winston Churchill.
Protesters have threatened to take the bank to court for discrimination.
Historical figures were first seen on British banknotes in 1970. Florence Nightingale is the only other woman to appear on a note.
The public can put forward suggestions on who should appear but the Bank said it only considers figures who have made a significant contribution to their field.
NBC reportedly holds celebs hostage to Jimmy Fallon's show
Ray Liotta sues skin care company over use of likeness