CBA chairman Catherine Livingston said the exact timing of Narev's departure would depend on when a replacement was found.
After AUSTRAC announced that it was filing a civil case in the Federal Court, Narev had said he should be the one to sort out the mess as it happened under his watch.
The board, which last week pledged that he had its full support, has apparently done a U-turn and decided that the third scandal under Narev will be his last.
Speculation has been rife ever since the AUSTRAC announcement that Narev had finally come to the end of his tenure.
He had ridden out two earlier scandals - one involving the bank's financial services arm and a second involving the banks life insurance section.
The bank had blamed coding errors in a software update for its Intelligent Deposit Machines for its allegedly falling foul of the law.
It claimed that when the IDMs were rolled out in May 2012, they were working as they should.
CBA added that a software update later that year led to the machines not providing the correct Threshold Transaction Reports which had to be sent to AUSTRAC. Any transfer of more than $10,000 has to be reported within 10 days.