A new poll suggests that Leo Varadkar is the favourite to lead Fine Gael.
Right now a third of voters want the Dublin West TD to take over from Enda Kenny - that's twice the popularity of the next choice Simon Coveney.
The third name offered to voters, by Behaviour and Attitudes researchers, was that of Justice Minister Frances Fitzgerald, but she has only 12% support.
The results from the poll, which was carried out between the 4th and 14th of this month, are published in today's Sunday Times.
The paper adds that Varadkar was widely commended earlier this year when he became the first Cabinet Minister to publicly reveal that he is gay.
