Richard Bruton and Simon Coveney are both being linked with the Health Minister's job in the upcoming cabinet reshuffle.
Government sources are suggesting the Taoiseach will want a "safe pair of hands" for the high profile portfolio.
Many challenges lie ahead for whoever gets the job, with overspending and medical cards among the key issues.
Transport and Tourism Minister Leo Varadkar has also been linked with the post.
Speaking as he attended the Irish Aviation Authority's Global ATM summit in Dublin this morning - he said the Taoiseach will be the one who chooses the make-up of his team: