The announcement came a day after Kenny said he would quit, kicking off a succession contest that was confirmed as a race between two younger ministers after other potential candidates stepped aside and colleagues began to pick sides.
Fianna Fail leader Micheal Martin said Kenny had worked diligently for his country over many years.
Dubliner Varadkar is now the clear favourite with the bookies to succeed Enda Kenny as Fine Gael boss, though political sources say it is way too early to call.
As to the future Cllr Mulhall said the party has "two good candidates" in Leo Varadkar and Simon Coveney and there was talk of a third candidate.
Her predecessor David Cameron said Mr Kenny "was a strong leader for Ireland, a great partner for the United Kingdom, and remains a good friend".
Minister and Donegal TD Joe McHugh has announced he is backing Social Protection Minister Leo Varadkar to be the next Taoiseach.
And he said he was happy with the campaign so far: "There's a long way to go yet...certainly looking forward to getting the campaign going around the country to the members and councillors".
Labour Party leader Brendan Howlin complimented Mr Kenny for bringing Ireland "back from the brink".
He told his party he would remain as leader for a "brief but appropriate period" to facilitate a leadership contest.
Junior minister Paul Kehoe also backed Mr Varadkar, Ireland's first openly gay cabinet member.
The other is Simon Coveney, the housing minister, who is the son of former Fine Gael minister Hugh Coveney.
"He also was part of reducing the Value-Added Tax rate for the hospitality sector, if you talk to any hotelier or anyone in the hospitality sector and they'll say that's one of the major reasons they can attract overseas visitors to Ireland".
The Fine Gael leadership will be decided by an electoral college that gives the parliamentary party - TDs, Senators, and MEPs (Members of the European Parliament) - 65 percent of the vote.
"He introduced the social welfare increases in the last budget and more importantly as an urban TD he introduced the social scheme for farmers and fishermen".
Simon Coveney is in Clare today for a number of engagements including the Clare Fine Gael AGM, but his visit here comes as he faces an uphill battle to defeat Leo Varadkar. Following this he was the Minister for Defence between 2014 and 2016.