The offer was a far cry from the 16 per cent increase teachers felt was needed to retain and recruit staff.
Rutherford said an electronic ballot could held at the beginning of next term over whether the strike should be extended for the whole day.
This is the first time teachers have taken industrial action since 1994.
In the meantime, negotiations between the NZEI and the MoE will continue as they try to reach a settlement.
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."
Today's announcement of industrial action by teachers comes just days after the country's nurses cancelled notice of nationwide strike action after receiving an improved offer from DHBs.
The New Zealand Nurses Organisation (NZNO) said it would vote on whether to accept the offer next week - when its first strike had been planned.