Huawei said on Wednesday that they were "looking into the situation" and promised to "actively address any concerns and work together to find a way forward".
Intelligence services minister Andrew Little told Reuters that Spark - whose request was part of the country's first 5G application - could work with the agency to mitigate risk.
We would never ban a particular company or a particular country.
Wellington also dismissed suggestions that its intelligence services came under pressure from allies in the "Five Eyes" spy network to bar Huawei, amid fears about cyber security and its potential links to Beijing. The decision was taken on the recommendations by the United States which said that the usage of the 5G equipment from Huawei makes the whole network vulnerable to snooping or interference.
The Minister responsible for the GCSB Andrew Little told RNZ on Thursday that the government is concerned with the equipment.
Spark said in a statement it would review the decision before deciding if further steps were needed and that, while disappointed, it was confident of meeting its 2020 network rollout target.
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He also denied there was any evidence of wrongdoing by Huawei. "We hope New Zealand will provide a level-playing field for Chinese enterprises' operation there and do something conducive for mutual trust and cooperation".
On Sunday TVNZ1 Q+A host Corin Dann questioned GCSB Minister Andrew Little about the decision New Zealand would have to make, after the Wall Street Journal reported the USA urged Five Eyes countries to avoid using Huawei.
Huawei has been involved in other telecommunications systems in New Zealand such as its 4G mobile network, and is investing NZ$400 million into research and development.
Little emphasised that the relevant security legislation is "project-based".
Spark said it had wanted to use Huawei 5G equipment in its planned Radio Access Network, which involves technology associated with cell tower infrastructure.
Little and intelligence officials have said that 5G networks have different security concerns to previous generations of mobile networks because it was hard to restrict vendors considered high risk to less sensitive parts of the network. "We believe that our involvement in the telecommunications sector has benefited New Zealand's economy, businesses and consumers".
He said the technology architecture for 5G is not that different to the other networks we already have.
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He also said that "another potential pregnancy" of a gene-edited embryo was in its early stages. The experiment has in no way been verified or published in any academic journal.