Ban Ki-moon said it was “high time” for a female head, after more than 70 years of the UN and eight male leaders.
Of the 11 candidates in the running to take up the post, five are women.
The secretary-general must be recommended by the 15 countries that make up the UN Security Council and elected by the 193 countries in the General Assembly.
A new secretary-general is expected to be announced in the autumn.
By tradition the role rotates between regions of the world.