The new Fine Gael leader has been meeting with independent ministers this week
Leo Varadkar says the Government is "going in the right direction" following a series of meetings with TDs and ministers.
Mr Varadkar has secured the abstention of Fianna Fáil, which should all but guarantee his election as Taoiseach in the Dáil next week.
However, the largest opposition party says it wants "confidence in the administration of justice" restored by a new cabinet.
The new Fine Gael leader met with Micheál Martin Wednesday afternoon, as part of a round of meetings to secure enough votes to replace Enda Kenny as Taoiseach.
Despite Minister Varadkar's election as party leader last week, Mr Kenny remains in the role until the Dáil confirms a successor.
Fianna Fáil's 'confidence and supply' agreement with Fine Gael means they will abstain in the election of a Taoiseach, with the deal set to be reviewed at the end of 2018.
Mr Martin told reporters this afternoon that he raised several issues with Minister Varadkar - especially in health and housing, where more needs to happen.
He explained: "I made it clear to the Minister [...] that the overarching objective of his government will have to be the restoration of confidence in the administration of justice and policing in this country."
Mr Varadkar met independent Ministers Katherine Zappone and Denis Naughten this afternoon, as well as Clare TD Dr Michael Harty.
Minister Naughten says they want greater delivery on the Programme for Government.
The Communications Minister argued: "We feel that the type of progress that should have taken place over the last 12 months hasn't. We want to see a renewed effort now to deliver on those commitments".
Minister Zappone had a list of items she wants addressed: "My thoughts are particularly with Ibrahim Halawa, his family and his advocates and supporters.
"I'm going to be discussing this with Minister Varadkar that I think we need to re-assess our approach to achieving his freedom".
In a statement, Mr Varadkar says: "I had a series of meetings today with Fianna Fáil Leader Micheál Martin, and Independent ministers and TDs.
"The meetings focused on legislation that needs to be prioritised, improving the working of the Dáil, and accelerating the implementation of commitments in the Programme for Government.
"All of the meetings were useful and I feel that we are going in the right direction."
Sinn Féin deputy leader Mary Lou McDonald, meanwhile, said she does not believe that "behind closed doors talks or deals are any substitute to a democratic mandate".
She added: "I think we need an election. I know that this Dáil has almost been like a slow bicycle race... it's just been turgid.
"It's the 'do something but not much' Dáil."
Additional reporting: Jack Quann