Updated: 13.10
The Government Chief Whip has said the Taoiseach should clarify when he will begin the Fine Gael leadership process.
An Irish Times/Ipsos MRBI poll published this morning asked 1,200 people who they like to see succeed Mr Kenny.
31% of respondents opted for Leo Varadkar - putting him ten points ahead of Simon Coveney.
Frances Fitzgerald was in third place on 10%, followed by Simon Harris (5%) and Paschal Donohoe (3%). 29% of respondents had no opinion, with the remaining 2% opting for other candidates.
The poll results were released following a turbulent week in the Dáil.
Speaking to Newstalk, Varadkar stated: "There is no vacancy at the moment. Everyone in this Government, including me, has a lot of work to do and my focus is firmly on the Department of Social Protection.
"The Taoiseach has made clear he will not lead the party in a future general election. This means there will be a leadership election at a future date. That will be the appropriate time to deal with the issue of the leadership of Fine Gael. The vacancy won't arise until the Taoiseach decides it should."
A number of Independent ministers yesterday voted against the rest of the Government to support Mick Wallace's Bill on abortions in cases of fatal foetal abnormalities.
Earlier in the week, Joe O'Toole announced his resignation as chairman of the expert commission on water.
Attention has now turned to Mr Kenny's future, and political reporter with the Irish Times, Sarah Bardon, says there are a number of prospective candidates up for consideration.
"It's been a very difficult week for the Taoiseach," she said.
"Fine Gael backbenchers are getting extremely anxious, and are calling on Mr Kenny to clarify when he intends to step down."
Regina Doherty has said there will be confusion until Enda Kenny outlines the process for choosing his successor.
Speaking to LMFM, the Chief Whip said the Taoiseach should provide some clarity as to when he will step down.
"I think it would be fair if the Taoiseach would put clarity on that so we could get back to normal," she argued.
Leo Varadkar yesterday said he would love to lead Fine Gael "whenever a vacancy arises".
The Minister for Social Protection says it is something he has wanted to do since he joined the party as a teenager.
Mr Kenny has previously indicated he will not lead his party into the next general election, but intends to serve his full term as Taoiseach.
Meanwhile, Simon Coveney says the issue of the leadership of Fine Gael will have to be looked at in the "not too distant future".
