Mr Varadkar will step down as party leader immediately and his resignation will take effect once a successor is elected.
Paschal Donohoe, Simon Harris, Simon Coveney and Heather Humphreys are among those being considered.
Mr Donohoe is minister for public expenditure and also president of the eurogroup of finance ministers in Brussels. Mr Coveney is a former deputy prime minister and foreign minister, who was deeply involved in Ireland’s diplomatic outreach during Brexit.
Thirty-seven-year-old Mr Harris is education minister, while Ms Humphreys is a minister for rural development, who is the only Presbyterian in the Irish parliament and has described herself as a protestant, Ulsterwoman and proud Irish republican.
Irish media reported that the change in leadership would not trigger an early election in a country struggling with a housing crisis and tensions over immigration but enjoying a healthy Budget surplus.
Mr Varadkar will be aware that his centre-Right Fine Gael and coalition partners Fianna Fail have long lagged far behind Left-wing Sinn Fein in the polls.
Ireland must hold its next general election, which Sinn Fein is expected to win, by March 2025.
In the last elections in 2020, Fine Gael finished behind Sinn Fein and Fianna Fail.
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