"Further significant technical issues have been identified in Huawei's engineering processes, leading to new risks in the United Kingdom telecommunications networks", read annual findings from the Huawei Cyber Security Evaluation Centre oversight board.
Tugendhat said it was hard to define core and non-core with 5G and that the Chinese company should not be allowed to build Britain's 5G network.
"What I see playing out here is a discussion amongst all of us about the realities of where do you define sensitive networks, where does that start and end".
This comes amid the leakage of last-ditch CIA warnings which were addressed to the British government and which claimed that Huawei had been funded by branches of China's military and intelligence.
"It isn't an open-and-shut case and it's important we review the advice properly and the Prime Minister and her closest advisers come to a decision based on that best quality advice".
May's Downing Street office declined to comment on the story.
A range of ministers spoke at the NSC meeting, with Philip Hammond, the chancellor, Greg Clark, the business secretary, and David Lidington, May's effective deputy, among those content to go along with a partial ban.
But the former USA homeland security adviser Tom Bossert said: "I think it is a little overly rosy and optimistic to suspect that [risks] can be mitigated in new 5G infrastructure ..."
A number of ministers, including Home Secretary Sajid Javid, Foreign Secretary Jeremy Hunt, Defence Secretary Gavin Williamson, International Trade Secretary Liam Fox and International Development Secretary Penny Mordaunt were said to have raised concerns about the decision.
Huawei is the leading manufacturer of equipment for next-generation 5G mobile networks with nearly instantaneous data transfer that will become the nervous system of Europe's economy, in strategic sectors like energy, transport, banking and health care.
Despite this, the British green light will doubtless be greeted with a sigh of relief at Huawei - and appears to foreshadow next month's report from UK.gov's telecoms infrastructure supply review, which Huawei recently told us it was concerned about.
Some critics have expressed concerns that the Chinese government could require the firm to install technological "back doors" to enable it spy on or disable Britain's communications network.
It is not clear to what extent the UK's decision may have been influenced by the United States, which has been placing pressure on other countries to follow its lead in cutting out Huawei from critical infrastructure.
A Department for Digital, Culture, Media and Sport spokeswoman said the security and resilience of the UK's telecoms networks is of "paramount importance".