The Green Party want its leader to share the position of Taoiseach if they form a coalition with Fianna Fáil and Fine Gael.
The Irish Times reports the party would want Eamon Ryan to lead such a government for one out of a five year term.
Fine Gael has said it would want the position of Taoiseach to be rotated in any coalition with Fianna Fail.
Micheál Martin says he would go into government with Leo Varadkar's party, and has not ruled out a rotating Taoiseach.