Walsh stepped down after a contract dispute with the IABA, saying it had "not made it possible for me to continue on in the role".
"I want a rethink within the next 24 hours by the IABA of their total approach to this," Irish Sports Council chairman Kieran Mulvey told RTE.
"We'll have to review our relationship with the IABA."
Walsh stepped down on Monday after 12 years in the post to take up a new position in the United States.
Ireland won Olympic, World and European gold medals at amateur level during Walsh's successful period as coach.
Mulvey added that a 22 August agreement with the Irish Amateur Boxing Association, relating to Walsh, was never put to its board.
"My view now is that deal should be offered to Billy even at this late date and give up the nonsense that's going on now in the IABA about this," he said.
"What was so unacceptable about the agreement of the 22nd August that it was never put to the board and never put to Billy Walsh even though it was agreed by him - what's the problem? I don't know what the problem is.
"Billy Walsh gave up in frustration - all you have to hear is his statement of a man who does not want to go America but is being forced by the petty indignities he was facing in this country."
The IABA said it had "done its utmost to retain Billy Walsh as Head Coach".
The Irish Sports Council says funding to the IABA will be reviewed after Ireland coach Billy Walsh resigned.