Update 10.45am: The Health Minister believes the dispute with nurses can be resolved but it will require some imagination.
A third day of strike action is underway with three more strikes planned for next week.
Thousands of medical appointments have been cancelled with the HSE warning it will become more difficult to reschedule patients if the action continues.
In the Dáil today, members of the opposition called on the Government enter talks without preconditions. A group of nurses looked on at the Dáil debate from the public gallery.
Minister Simon Harris thinks the dispute can be sorted out and he called for " further dialogue and engagement".
Mr Harris said: "It is essential that industrial action comes to a halt as soon as possible.
"Our Government is sincere in its offer to want to talk and engage with nursing unions to reach a resolution to this dispute.
"I am confident that this can be accomplished through further dialogue and engagement.
"I really hope after today's stoppage comes to an end that there is a real intensity of engagement as we work together to try and avoid further disruption."
Having faced rain and snow on the previous days, nurses in Cork were met with sunshine today - which has strengthened their resolve.
Dozens joined the picket at CUH to send the Government a message: pay up.
And despite the impact on the general public - some 50,000 patients have had appointments cancelled - there is no drop in support. Coffees, biscuits and fruit were hand delivered from supportive businesses and passing motorists continued to show their support.
Jean O'Connor, a nurse of more than 30 years, said her colleagues remain resolute in their stance.
"We are fighting for our profession, for our colleagues. We remain resolute and we won't back down," she said.
Meanwhile, General Secretary of the INMO, Phil Ní Sheaghdha, visited nurses and midwives outside the Coombe Women and Infants University Hospital today.
Ms Ní Sheaghdha said they have had no "correspondence" from the Government and members remain "resolute".
She said nurses and midwives on picket lines throughout the country understand that Ireland has now become a very uncompetitive area for recruitment and retention of nurses.
Ms Ní Sheaghdha said Ireland remains the lowest-paying country for nurses in the world. "And that has to be corrected," she said.
She was asked about the HSE's mounting concern about risk in the system - that the nurses' strike is putting patients at risk.
"What we are doing is highlighting the risk that exists in the health service every single day. There is a massive risk in the health service on a daily basis due to the staff shortages. Two days of a dispute have lifted the lid on that."
Ms Ní Sheaghdha said the HSE managers who were working the 24-hour cycle with nurses recognise the risk.
Ms Ní Sheaghdha said the nurses "linked in" with the HSE every two hours and will meet the health authority again this evening.
"We are satisfied that the emergency services we are providing during the days of the dispute are providing the level of care that is required."
By Vivienne Clarke and Digital Desk
The Minister for Communications, Climate Action and Environment, Richard Bruton, says that any solution to the nurses' strike must not have a knock-on effect on the remaining 400,000 public service workers.
He told RTE’s Morning Ireland that it is a difficult position to reconcile, but that the Government is committed and both sides need to get into the Workplace Relations Commission “to work on this.”
The first thing is to “get into an engagement process.” He said the Government is prepared to talk about pay in the context of the Public Service Pay Agreement that “binds everyone.”
Mr Bruton said issues such as rostering and productivity and other “cost neutral” changes can be made to resolve the dispute.
It is all a matter of balance and both the Minister for Health and the Minister for Finance and Public Expenditure were seeking creative solutions, he added.
In relation to the national children’s hospital, Mr Bruton said that the Government has to get to the bottom of why the protections they put in place to avoid overruns “haven’t happened.”
He said there was “no cheaper offer on the table” and the best international advice was that “we wouldn’t get it cheaper.”
“This is a very complex project, something went seriously wrong here. We are determined to take control of the situation.”
Meanwhile, the Taoiseach says efforts will be made over the coming days to find a mechanism to resolve the nurses' strike, which will escalate to three consecutive days of strikes next week.
Nurses working in Naas General Hospital have urged the government to engage in talks.
"Engage with us. We're not looking for the unreasonable," said one nurse. "We just want safe staffing levels."
Another nurse said: "We are our patients' advocate."
She said she wanted safe practice in all hospitals and "all we want you to do is to listen and cooperate with us."
"This has been going on for at least 30 years that I can remember," said another nurse. "Engage now. Deal with it now."
More than 25,000 medical appointments have been cancelled today, as nurses take to the picket lines for a third day.
Their dispute with the government shows no sign of ending, with both sides holding firm on the issue of pay.
Day three of the nurses strike gets underway at 8am and once more, more than 25,000 appointments including for outpatients and day surgeries, have been cancelled.
Injury units are closed, as are day centres for older people and those with disabilities, while community nursing and respite admissions will not take place.
The HSE says it is getting harder to reschedule the tens of thousands of appointments as the industrial action continues.
37,000 INMO members are taking to the picket lines in their row with the government over pay and staffing shortages.
They say addressing pay is the way to attract and keep nurses in Ireland.
The Government says pay increases for nurses are included in the public sector pay agreement and conceding to the nurses' demands would cost €300m.
Both sides last met at the Labour Court a week and a half ago. It decided they were too far apart to intervene.
No talks have taken place since.
The nurses plan three consecutive days of strikes next week.