And in a clear swipe at his rival Mr Varadkar, Mr Coveney said Mr Kenny should be "trusted" to announce his own timeline to step aside.
Also yesterday, Mr Harris listed the credentials he believes would make him an ideal candidate to take over as leader of Fine Gael, and dismissed any suggestion he would be too young to take charge if an opportunity arose.
Kenny, who has been prime minister since 2011, has said he will not lead the party into the next election due late next year.
Speaking today in Cork, the Housing Minister said the party must trust the Taoiseach to know what's best for the country.
"Let's wait and hear what he has to say on Wednesday evening".
"The business of people sitting at parliamentary party meetings tweeting out what's going on is absolutely corrosive".
Last night, one of the main contenders to succeed the Taoiseach, Minister Varadkar, called the current situation distracting and destabilising for the Government.
Flu widespread in 46 states; 9 more kids' deaths reported
Landers reminds the public that certain people with influenza are at higher risk of complications from the disease. Flu symptoms include fever, chills, cough, sore throat, runny nose, muscle or body aches, fatigue and a headache.
The move means it is very unlikely he will now face a no confidence motion from TDs or senators.
Declared contenders Simon Coveney and Leo Varadkar agree that a new leader should be appointed after the Taoiseach returns from his St Patrick's Day trip to the US.
CARGO TRAGEDY Aer Lingus to be sentenced for health and safety breaches after driver death at Dublin Airport...
Flanagan said his remark on a private Fine Gael group was "jovial Sunday afternoon banter", adding that he regrets that it has been taken so seriously.
Minister Flanagan, who holds the Foreign Affairs and Trade brief, messaged that Minister Harris wanted out of his role as minister for health.
He said: "It was a joke which should be taken as that.
I have full confidence in the Taoiseach to settle it", the statement said.