But the Taoiseach pledged to continue pressing the matter with British counterpart, David Cameron, saying it was the Dáil’s unanimous wish that there be such a probe.
He was speaking after Mr Finucane’s widow reacted with anger after Mr Cameron had proposed only a review of the case by a senior British lawyer.
Speaking after a meeting with the British prime minister in 10 Downing Street, Gerald Finucane said she felt “insulted” by Mr Cameron’s proposal of a QC-led review.
She said the family was “very disappointed” with the proposal and would not support the initiative.
The Finucanes want a full independent inquiry into the loyalist shooting in 1989. There have been persistent claims of security force collusion with the killers.
“I am so angry and so insulted by being brought to Downing Street … to hear what the prime minister had on offer,” said Ms Finucane.
“He is offering a review. He wants a QC to read the papers in my husband’s case and that is how he expects to reach the truth. All of us are very upset and very disappointed.”
The issue was raised subsequently in the Dáil by Sinn Féin leader Gerry Adams, who called on the Government to press Mr Cameron for a full inquiry.
Mr Adams said that the Finucane case was hugely significant in terms of indicating the levels of collusion that occurred during the Troubles.
He pointed to the recommendations of Canadian judge Peter Cory, who examined a series of killings involving claims of collusion at the request of the two governments. Judge Cory recommended that a full inquiry be carried out into the Finucane killing.
Mr Adams said Mr Cameron’s decision to ignore this recommendation proved that the British government was “playing the long game again”.
In response, Mr Kenny said he had spoken to Mr Cameron before the British prime minister’s meeting with the Finucane family.
Mr Kenny said he had reiterated that the Dáil had passed a unanimous motion calling for a full inquiry.
He said he told Mr Cameron that “if Geraldine Finucane was not happy with what was on offer, then clearly we would not be happy either”.
He admitted that he could not force the British government to launch an inquiry, but said he would continue pressing the matter with Mr Cameron.
He said Tánaiste Eamon Gilmore would meet shortly with Ms Finucane “to see where to go to from here”.
Mr Finucane was shot in front of his family in February 1989 by Ulster Defence Association gunmen.
He had represented people accused of republican terror offences in high-profile cases.