Among Mr Kenny's final official overseas engagements will be a two-day trade mission to Chicago and the centenary commemoration of the start of the World War One Battle of Messines.
Mr Kenny leaves the Fine Gael leadership as the party's most successful taoiseach.
He added: "Most importantly, I want to take the country forward".
Kenny, prime minister since 2011 and leader of his party for nearly 15 years, is expected to be replaced by either Leo Varadkar, the social protection minister, or Simon Coveney, the housing minister.
In a statement released after the meeting Mr Heydon said: "Under Enda Kenny as Taoiseach, we saw a devastated economy repaired, long ignored social issues addressed, and the rebirth of a self-confident Ireland".
"He would be a quite capable of leading the party and the other man Leo Varadkar would as well.but I would be sticking my neck out and saying I would be supporting Coveney", he said.
Earlier this week, the Irish Independent reported that if successful in the leadership contest, Varadkar will seek a special arrangement to keep Northern Ireland within the single market to ensure an "invisible" border between the north and the rest of Ireland.
Kenny's successor as party leader is set to be elected by June 2 and Noonan may still oversee the flotation of Allied Irish Banks, which he said could happen in the coming weeks, another landmark in the banking sector recovery.
"I'm really humbled, really grateful for that", he said.
"And also says that people should have an equal opportunity in this society to succeed". So who would you vote for between Coveney and Varadkar?
Cliona Doyle, chair of the Fine Gael parliamentary party, also pointed to Kenny's stewardship of the economy.
His supporters hope to narrow Mr Varadkar's lead, or pull ahead of him, before the series of hustings that will take place next week and mark the engagement between the candidates and rank-and-file members.
Michael Noonan has announced his intention to step down as Finance Minister and that he will not be seeking re-election at the next General Election.
Other outsiders include Justice Minister Frances Fitzgerald, Education Minister Richard Bruton, who launched a failed bid to oust Mr Kenny back in 2010.
Quoting our article, Cllr Doyle said: "I have a huge amount of respect for both Simon Coveney and Leo Varadkar".