The offer was a far cry from the 16 per cent increase teachers felt was needed to retain and recruit staff.
On Friday the union's national executive approved a secret online ballot of members to vote on extending the strike to a full day.
NZEI Te Riu Roa president Lynda Stuart said over the past few weeks their members told them of their resolve to send the strongest possible message to the Government about the crisis facing schools.
"We have listened to members and will now ask them to vote by secret ballot to extend the proposed three-hour strike to a full day," Stuart said.
"It is obvious that teachers are committed to working for change to ensure all children get quality teaching and learning.
"They are saying that it's time for them to be valued for the professionals they are, that we need to attract and retain teachers, have more time to teach and to have truly inclusive schools where all children can learn and thrive."
The ballot would run for a week from this Wednesday, July 25.
This would be the first time teachers had taken industrial action since 1994.
Earlier this month Ministry of Education deputy secretary of early learning and student achievement Ellen MacGregor-Reid said she was disappointed the offer had been rejected while negotiations were still happening.
"The offer sees trained teachers' base salaries increase between 6.1 to 14.7 per cent over three years which would see the beginner teacher rate increase to $50,280 a year, rising to $55,030 in 2020.
"The offer would also see increases for principals of between 6-11 per cent which would see the starting base remuneration of a principal of a school of 50 or fewer students increase to $92,873 in 2020."
National education spokesperson Nikki Kaye said the Government needed to handle its negotiations better to avoid disruption to childrens' learning and it had no excuse not to deliver on its promise.
"This is simply a case of Labour prioritising tertiary students over primary school teachers and students. It can't say there's no money left when it chose to spend $2.8 billion on a fees-free policy that saw next to no increase in university participation."