The Green Party wants to fill the role of Attorney General in the next government - or have a "neutral" candidate take the position.
There are fears among Greens the role will be politicised if held by Fianna Fáil or Fine Gael and used to block significant legislative changes.
Senior figures in the Greens want the party to select the next Attorney General or ensure the person selected for the role is not linked to either party with which they are negotiating.
There is also a push within the party to secure a ministry of state in the Department of Public Expenditure and Reform or Department of Finance.
The Greens want to make sure they have as much influence as possible over decision- making in government.
Party sources believe it is essential the party has oversight of spending and legislative decisions if it is to join Fine Gael and Fianna Fáil in power.
A Green or neutral Attorney General would ensure the other two parties could not use the office to block legislation.
Current Attorney General Séamus Woulfe is long-time Fine Gael and a long-time supporter of Communications Minister Richard Bruton.
Máire Whelan was appointed Attorney General in 2011 by the Fine Gael and Labour Party coalition government. She was a former Labour Party financial secretary.
Former Taoiseach Enda Kenny reappointed Ms Whelan when he was re-elected Taoiseach in 2016, despite widespread speculation she would be replaced due to the fall-out from the Garda Maurice McCabe scandal.
Meanwhile, Fine Gael has been pushing for controversial legislation to overhaul how judges are appointed to be included in the programme for government.
The outgoing government failed to pass the Judicial Appointments Commission Bill, which was championed by outgoing Transport Minister Shane Ross, after it was filibustered in the Seanad for more than a year.
The bill automatically lapsed when the General Election was called in January.
Despite opposition from some in the party, Fine Gael has pushed for its inclusion in a new programme for government in the three-party talks in a move which has surprised Fianna Fáil and Green Party negotiators.
Fine Gael included a commitment to pass the bill in its manifesto, where it committed to reform the way in which judges are appointed "by the passage of the Judicial Appointments Commission Bill".
The party has continued to argue for the bill in the government talks despite Fianna Fáil's trenchant opposition to the legislation.
The issue is understood to have been parked and will be subject of further discussions.
One source said that while it was unlikely to be a deal-breaker, the new programme for government may include a commitment to reform judicial appointments with new proposals different from Mr Ross's.
The bill supported by Mr Ross proposed setting up a new body, chaired by a non-lawyer and with a non- legal majority, which would advise the government on judicial appointments. It would be able to propose a maximum of three names to the government for each judicial vacancy.
It was strongly opposed by many in the legal profession, Fianna Fáil and many in Fine Gael, including a majority of the party's senators.
The outgoing Fine Gael-led administration committed to its passage in return for Mr Ross's continuance in government.
Justice Minister Charlie Flanagan sat through more than 110 hours of debate in the Seanad up to last December prior to the General Election being called, which resulted in the bill lapsing.
One of the most controversial elements of the previous bill was the idea that it would have a lay chair and would not be chaired by a member of the judiciary such as the chief justice of the Supreme Court.
Yesterday, government formation talks centred on farming and agriculture.
Fianna Fáil and Fine Gael fear alienating their rural bases by capitulating to the demands of the Green Party on climate change.
However, the Greens are also increasingly aware of the issues facing rural communities and are eager to put forward policies which will help farmers diversify. The party has suggested new grants for farmers if they use sections of their land for biodiversity.
Those seeking to remove Eamon Ryan as Green Party leader have raised concerns about his lack of rural credentials.
Green Party deputy leader Catherine Martin, who is originally from Monaghan, is being touted as an ideal candidate to replace Mr Ryan because of her rural background.