Negotiating teams from Fine Gael and Fianna Fáil held what were described as "constructive" talks on Friday afternoon.
Both sides have agreed to meet again on Monday.
“The negotiating teams for both Fianna Fáil and Fine Gael met again this afternoon. It was a constructive meeting. Contacts between both parties will continue over the weekend and it was agreed that they will both meet again on Monday,” said both parties in identical statements.
Earlier, Minister for Agriculture Simon Coveney has said a new government could be in place by the middle of next week.
“We need to get on with it, we need to work out what Fianna Fáil needs in order to do that. If both sides approach this with a view of doing what’s best for the country, then we can have a government by the middle of next week.”
He said that ‘pragmatic discussions’ were ongoing with Fianna Fáil and he expected that parallel discussions will be held with the Independents and smaller parties.
Speaking on RTE’s Sean O’Rourke Show, he said it would be better if there was a written agreement between the parties.
“We will need clarity on what is agreed. There will have to be clear ground rules for week to week operations.
“It is sensible and reasonable to want an agreed timescale.”
Fianna Fáil negotiator, Offaly TD Barry Cowen, speaking on the same programme, said that next Wednesday was possible but that his party would not be guided by a deadline.
“It is up to Fine Gael to build a government and the Independents will have to buy into it. The Independents can’t be in and out of government.”
Independent TD Finian McGrath told Sean O’Rourke that if the Independent Alliance was given an undertaking that their policies would be implemented, then they would support the minority government.
“The days of the big parties putting a gun to the head of the other parties, are gone.”
He said that at least one third of Cabinet places should go to non-Fine Gael TDs.
On Thursday, Fianna Fáil pulled out of the contest to form a minority government. In the coming days Fine Gael will attempt to assemble the numbers to have Enda Kenny elected taoiseach at the fourth attempt next Wednesday, following yet another unsuccessful series of votes yesterday.
Minister for Transport Paschal Donohue said a written agreement with Fianna Fáil is "an absolute must" in forming a minority government.
"We are looking for the ability to govern and to do that we will need a written agreement," he told RTE's Morning Ireland on Friday.
Clare Fianna Fáil TD Timmy Dooley said there were mixed views within the party on the issue of a written agreement. He said the party was looking at a broad based mechanism to facilitate a minority government.
Fianna Fáil would not participate in government but was “big enough” to show maturity and to look at facilitating a minority government, he told RTÉ
Fianna Fáil negotiator in the talks with Fine Gael, Dublin South TD Jim O'Callaghan said on Friday he norm with minority governments around the world was that a short document is drawn up about the fundamentals.
“We need more collaborative politics in this country. The country needs a government. It is not tenable not to have a government,” he told Newstalk Breakfast.
“Fianna Fáil are being responsible, but politics will have to be different now.
He said two weeks was the limit for forming a government.” It can’t go on longer than that.”
The Labour Party on Friday accused Fianna Fail of failing to act responsibly in its approach to the formation of government.
A party spokesman said Labour Party made it clear immediately after the election that it intended to go into opposition and the reason Fine Gael had turned to it for support at this stage was that Fianna Fail was refusing to cooperate in providing stable government.
“The only reason that, seven weeks after the election, Fine Gael has turned to Labour seeking some kind of support is that Fianna Fáil has shown its true colours. Fianna Fáil won’t go into government with Fine Gael, won’t encourage Independents to go into government with Fine Gael, and is determined to ensure an unstable minority government so that Fianna Fáil can trigger an election at a time of its suiting.
“This naked self-interest shouldn’t surprise anybody given Fianna Fáil’s record; having destroyed the economy in 2011, they are now determined to destroy any prospect of stable government at a time when the country urgently needs it. Micheál Martin talks of the need for reform when the truth is Fianna Fáil itself hasn’t remotely reformed at all - the party comes first, the people a distant second,” he said.
Red line issues
Longford-Westmeath Fine Gael TD Peter Burke said there can be no red line issues between Fine Gael and Fianna Fáil in terms of ironing out a solution for government.
Speaking on his local radio station Midlands 103 he said: “I am very hopeful that in the coming week that huge efforts will be made by Fine Gael to ensure that a stable government is put together.”Mr Burke said all parties must work together to ensure a taoiseach is elected.
The Dáil will reconvene on Wednesday next to elect a new leader, however Fianna Fáil says it will not put Mr Martin forward for the position for a fourth time.
Differences over Irish Water could scupper the chances of a Fine Gael-led minority government getting off the ground.
If the two parties cannot reach agreement on water and the rules to enable a minority government to function, another general election in May becomes a real possibility.
As well as talking to Independents in the coming days, Fine Gael will seek to persuade the Labour Party and the Greens to vote for Mr Kenny and participate in a minority government.
However, these groupings will be influenced by the progress of talks between Fine Gael and Fianna Fáil about how the operation of a minority government can be facilitated.
The attempt by Micheál Martin to force the hand of the Independents y backfired when 14 of them came together and agreed to abstain on the Dáil vote for taoiseach.
His nomination was defeated by 91 votes to 43, a margin of 48. While Mr Kenny was also defeated, the margin was considerably smaller at 25.
Fianna Fáil indicated late yesterday that Fine Gael must prove it has the declared support of at least an additional six TDs before Mr Martin agrees to facilitate the election of Mr Kenny as taoiseach next week.