The Fine Gael leader will lobby for support to hold a referendum on the Irish Constitution, which now gives the unborn and the mother an equal right to life under Article 40.3.3, The Irish Times reports.
The country will vote on whether to repeal the amendment and replace it with a provision to allow the Oireachtas to legislate for abortion.
Mr Varadkar is due to announce details on the Government's position on the referendum after a Cabinet meeting in Dublin on Monday evening.
The Taoiseach said his views had "evolved" since 2014 when he described himself as "pro-life" while acknowledging that current laws were too restrictive.
The law was changed three decades later to allow terminations when the mother's life is at risk, following public outrage at the death of a pregnant woman in 2012 who was refused an abortion. "As Minister for Health I became convinced abortion had no place in the Constitution".
Speaking on RTÉ's The Week in Politics, Mr O'Donovan said that while he believes the referendum should go ahead, he will not support a repeal vote.
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Ministers are expected to formally agree to hold a referendum on the issue.
Last December, a report by a specially convened parliamentary committee found that the section of the state's constitution that confers equal rights on the mother and unborn child - the Eighth Amendment - was not fit for goal and should be repealed.
That followed recommendations from members of Ireland's Citizens' Assembly to liberalise the law on terminations. "I'll be campaigning for them to be changed".
The committee also recommended abortion be available up to 12 weeks of pregnancy without a woman having to explain her decision.
The Minister of State, Patrick O'Donovan, has today said that he will not support the proposed repeal of the Eighth Amendment.