Several independent TDs have been told that they will be included in government formation talks the week after next.
It is understood that contact was made late last night with two of the groups of independent TDs that have been formed in the Dáil.
The call sought to reassure them that they are still in the reckoning to be part of the next government.
Fine Gael deputy leader Simon Coveney called the Independent Group and the Regional Group to tell them that negotiations would take place with them during the first week in June.
The three party leaders are expected to make contact with the independent groups next week with a view to more detailed talks taking place soon after.
All parties are understood to be working towards concluding the programme for government talks process by Sunday 7 June at the latest.
Should the Greens, Fianna Fáil and Fine Gael agree a programme for government, it would give a new government a majority of four, with the backing of 84 TDs.
However, there is believed to be an ambition within some parties to get the total number of TDs supporting the next government close to 90 seats.
This is in order to ensure that the new administration is durable and capable of making difficult decisions.
It is not clear how much input independent TDs will get to make into policy areas when they talk to the other parties.
There remains, however, the outside possibility that some independents could get a ministerial position.
Any agreement with independents would be expected to guarantee ongoing access to government ministers.
Michael Fitzmaurice, Marian Harkin and Michael McNamara of the Independent Group spoke to Fianna Fáil and Fine Gael a number of weeks ago.
While Seán Canney, Denis Naughten, Michael Lowry, Matt Shanahan, Cathal Berry, Verona Murphy, Noel Grealish and Peter Fitzpatrick of the Regional Group have also held talk with the parties.
The group's other member, Aontú leader Peadar Tóibín, has ruled out participating in any further talks with Fianna Fáil and Fine Gael.
Meanwhile, the issue of health - and in particular Covid-19 - is due to take centre-stage in today's talks between Fianna Fáil, Fine Gael, and the Green Party.
Speaking on RTÉ's Late Late Show last night, Fianna Fáil leader Micheál Martin said he believed a programme for government could be signed-off if "not by the end of next week [then] very shortly after that".
He said it would be an appalling indictment on politics if a government could not be formed.
This afternoon, Green Party TD Brian Leddin said the talks are progressing well.
Speaking on RTÉ's Saturday with Cormac Ó hEadhra, the Limerick TD said that the talks on energy policy this week went well and that both Fine Gael and Fianna Fail were ambitious about the agenda.
He said a range of measures needed to deliver a 7% reduction in carbon emissions are being thrashed out and the challenging areas of transport and agriculture will be considered this week.
On a potential leadership challenge in his party, Mr Leddin said that Eamon Ryan is the most successful party leader the Greens have ever had, and it does not make sense to him to have a leadership contest at present.
He said he has "huge regard" for deputy leader Catherin Martin, but said this is not the time for a leadership election.
Additional reporting Paul Cunningham