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Walker admitted to Muller on Monday midday he was behind the private details of active Covid-19 cases being leaked to media, sparking a Government inquiry.
Muller told the Clutha-Southland MP he needed to own up publicly.
It's understood later that afternoon - after the Government announced the inquiry - Muller received a legal letter on Walker's behalf.
It asked the National Party leadership not to out Walker citing concerns about his privacy.
Muller then sought his own legal clarification.
The National Party leader referred to this legal exchange with Walker in interviews this morning.
He told the AM Show: "I received a notification lunchtime Monday then asked Hamish to connect with my chief press secretary and chief of staff.
"And then, of course, we found out the inquiry was happening and it became very clear to me, very clear, that we needed to inform that inquiry of what we knew.
"I then shared that expectation with Hamish and then I received legal representation, which of course put me in a position of needing to check my own legal position before I made it very clear that that information needed to become public.
"That roughly that took 24 hours - it is what it is, that's how it happened. It was never going to be moved from the course that this information needed to be made public and it was."
A spokeswoman said given an inquiry had been announced and it was National's expectation that Walker "would be co-operating with that inquiry, both sides sought legal advice over what we could and could not say publicly, so that it didn't undermine the inquiry.
"And it took some to-ing and fro-ing before that came through."
The National Party leadership did not know former party president Michelle Boag was Walker's source until late Tuesday afternoon - hours before she outed herself in a press release.
Walker leaked the details of the patients to media as he believed by naming them, he would back up a claim the active cases were from "India, Pakistan and Korea" as he'd said in an earlier statement that was slammed as racist.
In his confession, Walker claimed he did it to "expose the Government's shortcomings so they could be rectified".
He said he'd sent the document to show the information wasn't password-protected or stored on a secure system which only authorised people could access.
After learning of Walker's role in the scandal, Muller immediately stripped the MP of his portfolios and wrote to the National Party board asking for him to be removed from the party.
It was also his "personal view" was Boag should step aside from having any involvement with the party.
Last night, Boag stood down from her role as acting chief executive of the Auckland Rescue Helicopter Trust - the role in which she'd received the patient's information.
She also today stood down from Auckland Central and deputy leader Nikki Kaye's campaign.
Boag said last night she'd made a "massive error of judgement on my part" and apologised for doing so.
"I very much regret my actions and did not anticipate that Hamish would choose to send it on to some media outlets but I am grateful that the media involved have chosen not to publish the 18 names that were contained within it."