Mr Varadkar, a doctor from Dublin and the favourite for the job, is Ireland's first openly gay member of cabinet.
Mr Kenny delayed his resignation on a number of occasions this year.
Candidates will have until Saturday to get eight parliamentary party members to submit their nominations, and endorsements are expected to starting coming immediately. It is understood she is contacting female representatives of the party to ensure her place on the ticket. However Justice Minister Frances Fitzgerald has not yet ruled herself out of contention.
Kenny will remain as a party leader in acting capacity until his successor is elected.
The contest is decided by an electoral college that gives the parliamentary party, made up of TDs, senators and MEPs, 65% of the vote.
Mr Coveney has the confirmed support of seven TDs so far while Mr Varadkar has 15. Others are undecided or have yet to declare.
He is the first Fine Gael leader to win re-election to office.
The party has said this lunchtime that the appointment of a new leader, in and of itself, will not require any amendment to the Confidence and Supply Arrangement with Fine Gael. His retirement as leader was effective from midnight on Wednesday.
Fianna Fail leader Micheal Martin said Kenny had worked diligently for his country over many years.
Questions will immediately turn to a successor and Ireland's next taoiseach alongside the analysis of Mr Kenny's legacy after leading the country out of a crippling recession, albeit with an agenda of austerity.
Enda Kenny announced Wednesday he will step down as Ireland's prime minister after his Fine Gael party elects a new leader.
Kenny received a standing ovation from his colleagues and was described as "very emotional" by those present.
Politicians in all parties paid tribute to Mr Kenny last night.