Mr Key said the situation had changed now that Mr Abbott was no longer in charge.
"Once the change of prime ministership happened, we wouldn't it consider it a binding obligation," he said.
The Australian Government's change of position was down to two factors, according to the Australian.
Mr Abbott did not consult his Foreign Minister Julie Bishop about the agreement. And former Prime Minister Kevin Rudd had expressed an interest in the UN position.
Mr Key said he was unaware that Ms Bishop had not been informed of the commitment.
He still hoped that Australia would back Ms Clark if she ran as a candidate.
"She's got all the credentials, she's extremely capable and personally [and] frankly I hope she puts her name forward and gets the job.
"I'm more than happy to continue to make the case as I would to any other leader I could talk to if she put her name forward."
However, he did not expect Australia to back Ms Clark over Mr Rudd if both were competing for the role.
Mr Key did not know whether New Zealand would support Mr Rudd if Ms Clark was not in the contest, saying it would have to be discussed with Foreign Affairs Minister Murray McCully.
UN Secretary-General Ban Ki-Moon's term finishes this year.