The offer included a one-off payment of $2000, a 2 percent increase in staffing, as well as two new steps on the pay scale.
The new steps would mean registered nurses with more than five years of experience could have their base salaries boosted from $66,755 to $77,386 by December next year.
But nurses say the benefits have not been spread across the sector.
The two proposed 24-hour strikes are slated for 5 and 12 July, with urgent talks between the New Zealand Nurses Organisation and DHBs to be held on Monday to resolve the dispute.
The government has committed to funding an extra 500 nurses - a two percent increase in nursing numbers.
Nurses Organisation industrial services manager Cee Payne and DHB spokesperson Helen Mason were both on Morning Report this morning laying out their case.
Ms Payne has been at the nurses union since 2005.
She was awarded a Harkness Fellowship in 2014-15, and has an MBA from the University of Strathclyde in Scotland and a nursing and midwifery degree from the College of Nursing and Midwifery at Addington Hospital in Durban, South Africa.
DHB and union representatives face off on Morning Report
On the latest offer and the rejection
"The difficulty with the current offer is that it ... leaves a lot of our members with 3 percent per annum over three years offer from the DHBs.
"And that has created distortions for our senior nurses, our community health nurses, our enrolled nurses in hospital aid."
"That offer we have made is already twice our previous offer."
"We believe this is a very good deal with an increase for our most experienced nurses, five years or more, over the course of this agreement they will have a $10,000 increase in their base pay rates if they are working full time.
"But we think there are lots of things in the package which are really important to nurses outside of base pay rates. So that includes the focus in the safe staffing programme as well as resources to ensure that we actually implement care capacity demand management (CCDM).
"I think our problem with CCDM is that it will take more time, it's an acute staffing crisis we've got.
"It's about time we got more nurses on the floor."
Who sets terms over money?
"With the first offer ... DHBs had gone as far as they could go and the ministry came in and supported DHBs to make an increased offer with this latest offer.
"The government sets the fiscal envelope for New Zealand ... in terms of health, at the end of the day."
"I believe that we will require additional funding to reach a settlement."
Is there more money available for nurses?
"Not with this offer.
"I don't think that the offer now ... is about more money, I think it's about nurses having trust and confidence that DHBs are going to fully implement care capacity demand management, recruit ... nurses, that's what I think nurses are really wanting to know."
Where to from here?
"We have agreed to mediation with NZNO ... and so whilst we are planning for the worst I'm still optimistic that through the mediation process that we'll be able to find resolution and DHBs are going to be doing everything we possible can to reach settlement with NZNO.
"If industrial action does go ahead we will need to defer and reschedule non-essential services. Emergency and essential services will be available."
Will the public back nurses if there are strikes?
"I believe we've got public support.
"Nurses wouldn't be doing this if they didn't genuinely believe that every day the expectations on them are beyond their ability to deliver safe patient care.
"If we can resolve this dispute before a strike that is what all the parties want, but it won't be achieved if we've got employer and government parties telling us what they won't do."