Early on Saturday, a covert meeting was held at the home of Jim O'Callaghan, the newly elected Fianna Fáil TD for Dublin Bay South. It involved only two political figures, O'Callaghan himself and Fine Gael TD and acting Minister for Health Leo Varadkar. The only others who attended were Enda Kenny's policy adviser Andrew McDowell and Micheál Martin's chief adviser Deirdre Gillane.

The meeting ran for over an hour. It didn't dwell on any policy detail or any red lines, rather with practicalities and modalities. Both sides quickly accepted that partnership was off the table.

"We all acknowledged that a minority government was the only show in town," a person familiar with the meeting said. While Fianna Fáil won't say it publicly, most of its big hitters tacitly accept that Fine Gael has its nose well in front on that score.

The mood of the meeting was cordial and realistic. Neither side wanted an election; both knew it was coming, the whole process was slowly grinding to an end. They would now have to work towards an accommodation that would enable a minority government to be formed.

Crucially, the two TDs agreed that formal talks between negotiating teams from both parties would

get under way early this week. After last week’s sulphuric exchanges between Kenny and Martin, this “back channel” represented a breakthrough.

The two big parties knew something had to be done after last week’s testy exchanges. The impetus may have come from the demand on Friday by the Independent Alliance for separate meetings with Kenny and Martin within 24 hours. That request was spurned but the big parties then took the initiative themselves.

The initial contacts were followed by careful and conciliatory public comments from Fine Gael and Fianna Fáil. This was particularly evident during Martin’s speech at the Liam Lynch commemoration speech in Co Tipperary on Sunday when he referred to the need to find a new model of government.

"We are offering a major compromise. We are not refusing to change. We will agree to a process which can allow a government to be formed and for that government to have reasonable security based on a fully transparent framework."

It is likely the negotiations teams (four on each side) will hold initial talks on Monday night in a neutral venue. The next vote for taoiseach is on Thursday but is not likely to produce a result.

So what’s at issue? Well, there is broad agreement the optimum solution will be a deal that lasts for 2½ years (or three budgets). The agreement will need to be written, although there are differences on detail, though.

Fianna Fáil will be reluctant to give any commitment on policy but will agree to commit on process – “confidence and supply” support, as it is known. It will offer support for no-confidence votes and on all financial and economic votes, provided deep reforms have been put in place. Therefore, fundamental parliamentary reforms will need to be agreed that will give a specific all-party budget committee specific powers to start discussing the budget months in advance. It will be an economic management committee, but one that extends to all of parliament rather than to the two government parties.

Irish Water is a potential stumbling block but Fine Gael sources are confidence that the gap between it and Fianna Fáíl has been “over-hyped”.

Fianna Fáil has talked in terms of a framework document, although Fine Gael and the Independents want more meat on the bone. All agree it is gettingclose to decision time. “We can’t have too many more Thursdays where votes are taken on the taoiseach,” said Darragh O’Brien of Fianna Fáil. “We can’t have a Groundhog Day situation developing.”

As of now, the most likely date seems to be Wednesday, April 20th. The shape? A Fine Gael-led minority coalition with a writ for 2½ years.