Talks between Fianna Fáil and Fine Gael to form a government are set to resume later on Monday after the Christmas break.
In November's general election, Fianna Fáil won the most seats at 48, but 88 are needed to secure a majority in the Dáil (the lower house of Irish parliament).
Even with Fine Gael's 38 seats, the parties will also need independent TDs (MPs) to support the coalition.
Both parties have previously ruled out forming a coalition government with Sinn Féin, the second largest party, who won 39 seats in November's poll.
Irish broadcaster RTÉ reports that both Fianna Fáil and Fine Gael are hoping to have a Programme for Government ready for 22 January, when politicians are set to elect the next taoiseach (Irish prime minister).
Negotiation teams – headed by Fianna Fáil's Jack Chambers and Fine Gael's Helen McEntee – are set to discuss transport, health, and climate policy at Government Buildings this week.
Last month, party leaders Micheál Martin and Simon Harris described a meeting about the possible formation of a government as a "positive, constructive engagement".
Sinn Féin president Mary Lou McDonald previously said it was "bad practice" to leave the party out of talks.
The last Irish government was a coalition of Fianna Fáil, Fine Gael and the Green Party, however the latter saw only one of its previous 12 TDs returned.
The position of taoiseach was rotated between Fianna Fáil and Fine Gael.
In Northern Ireland, a number of yellow weather warnings for ice and snow have been issued.
A 31-year-old has been charged with the murder of his mother in County Westmeath.
The body of Mirjana Pap, who was in her 50s, was found on lands at Curraghaleen on Monday.
John O'Connor, 58, was pronounced dead in Cluain Fada, New Ross, shortly after 11:00 on Tuesday.
Gardaí (Irish police) and emergency services were called to an incident at a residence in New Ross on Tuesday.
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