Michael Noonan could give the green light to the flotation of AIB during his final days as Minister for Finance this week.
The sale of existing shares in the bank, is set to be the biggest flotation on the Dublin and London stock markets this year.
The sale which equates to about a quarter of the AIB shares should raise between €2.5bn and €3bn.
But the Labour Party Leader Brendan Howlin wants Independent Ministers to block the move.
Howlin says the decision should be left to the new cabinet under a new Taoiseach after seeking the approval of Dail Eireann.