The Taoiseach and Tanaiste are expected to meet with representatives of the Magdalene women this week before they finalise their position.
But, speaking to the Sunday Independent, Social Protection Minister Joan Burton said she was "very confident an appropriate apology and a response incorporating a scheme to look after these women will be established".
She added: "A very great wrong was done to these women who are at the centre of this issue and I expect all my colleagues are united in this view."
In an indication of a shift within the Cabinet, another minister is believed to have admitted: "The men around the cabinet table made the wrong decision."
The Sunday Independent has also learned that sharp exchanges occurred between Enda Kenny and Labour junior health minister Kathleen Lynch during the private cabinet briefing about the Magdalene report prior to Mr Kenny's non-apology.
"To those residents who went into the Magdalene laundries from a variety of ways, 26 per cent from State involvement, I'm sorry for those people that they lived in that kind of environment," Mr Kenny had said in the Dail.
It is believed Ms Lynch became deeply unhappy with the Taoiseach's "wishy washy" approach to the matter at the briefing.
Ms Lynch had campaigned strongly for the rights of the Magdalene women in
One source told the Sunday Independent: "Kathleen buttonholed Enda in the way she can, and let him know in no uncertain terms that 'there will be an apology' and that 'it was not a case of if you give an apology but when'."
Mr Kenny, who is believed to have been "hugely cautious" over the "financial consequences of an apology", was also told by Ms Lynch that "you would be better off being seen to apologise of your own volition rather than having it forced out of you''.
Another source noted "it is widely known Deputy Lynch used very direct language to the Taoiseach at the meeting".
Ms Lynch, who is seen as being a strong Gilmore and government loyalist, was visibly uncomfortable when asked to defend the Government's position last week.
The controversy spilled over into a Labour parliamentary party meeting where a "mass walkout to the plinth by party members on behalf of the Magdalene Women was narrowly averted''.
Sources from within the party said party leader and Tanaiste Eamon Gilmore was "visibly taken aback by those leading the criticism of the governments 'apology' and the vehemence of the criticism".
Eventually the tension was diffused by the compromise where the party issued a collective press statement calling for "a formal apology by the Government" and the "provision of services for survivors".
The statement noted that "the parliamentary party was today united in the view that these women should be vindicated and honoured".