Ms Kennedy has been editor of the newspaper since October 2002.
Her contract was due to expire later this year when she turns 60 and she has announced she will not be seeking an extension.
It is understood she informed staff of her decision to step down in the paper’s offices in Dublin.
Geraldine Kennedy will remain on as editor until the process of selecting her successor has been finalised.
A native of Carrick-on- Suir in Co Tipperary, Ms Kennedy is a graduate of the College of Commerce, Rathmines.
Having worked for papers, including the then Cork Examiner, she became political correspondent at the Sunday Tribune and was one of the journalists whose phones were illegally tapped by the Irish Government in the 1980s.
She is also a former Progressive Democrat TD, but lost her seat in the 1989 General Election.
Later she became political editor of the Irish Times and when she took over from previous Irish Times editor Conor Brady, she became the first woman to edit an Irish national newspaper.
A number of names are likely to be in contention for the post — one of the highest paid in the Irish media.
Deputy editor Paul O’Neill is likely to be a candidate for the position, as is news editor Kevin O’Sullivan. Another person who could be considered is Hugh Linehan, the newspaper’s online editor.
The current Irish Times business editor John McManus is another who may be a candidate to fill the editor’s chair.
Other candidates could include the newspaper’s foreign editor Denis Staunton and assistant editor Fintan O’Toole.