The ministry said today that it appreciated the level of public interest in the case, but would not release any information on privacy grounds.
"As stated by Mr Peters, we have been in contact and the matter has been resolved to the Ministry's satisfaction."
Speaking to Newstalk ZB this morning, Peters said that he declared that he was in a de facto relationship with partner Jan Trotman when he applied for superannuation.
There appeared to have been an alteration to his application form but "no one knows how it was made", he said.
That meant he ended up getting the single person's superannuation rate, Newsroom reported, which was $60 more a week than the payments for a person in a de facto relationship.
It was not until Trotman applied for her pension that MSD discovered Peters' payments were too high.
Peters said he never realised the error because his payments were well below that amount which his married friends were getting.
"Frankly the figure was not high at all and way lower than most of the ones that I was aware of from people I knew who were married or widowed."
MSD had made it clear that there was no wrongdoing, Peters said. They sent him a letter thanking him for his quick co-operation.
Asked whether he would release this letter, and other correspondence with MSD, he initially said he would not.
"If you think I'm lying then that's too bad."
But Peters later said he would consider it.
"I hadn't thought about it, perhaps I should now when I reflect upon it."
Prime Minister Bill English said today the matter was for Peters, including if he should release records.
"Clearly there are questions there. And I think people are going to be asking. But it's up to him to deal with that...clearly there are more questions going to be answered.
"When you are a public person you have to pay attention to your own personal circumstances and how you deal with them because at any time they could come under scrutiny. People can make mistakes, we're all human, it's just a matter of what the particular circumstances are."
English said he would be concerned if someone working for MSD or IRD had leaked details about Peters' personal circumstances.
"People expect them to be treated confidentially."
Asked if the overpayment was an administrative mistake was he worried more New Zealanders could be getting too much super, English said the circumstances in the case weren't yet known.
"MSD have a system where they endeavour to be as accurate as possible...people's circumstances change...I wouldn't want to generalise from one case."
English said he wouldn't see any need to speak to Peters about the overpayments.
"That looks like it will be worked out through the next few days."
Labour leader Jacinda Ardern refused to get involved in the issue, saying it was a matter for Peters.
"It would be helpful if we had more information, but still, ultimately, this is an issue for Mr Peters to be answering questions on."
She would not be asking him for an explanation or more information.
Asked whether Peters should release his correspondence, Ardern said "transparency is always helpful" while adding "that's a question for Mr Peters".
Speaking to the Herald last night, Peters said "nobody knows why" he was mistakenly overpaid.
"You have a whole range, maybe six or seven ranges of payments for super. And I'd have been on one of the very lowest ones, and I think I was one up from where I should have been."
The matter was complex and he only knew the rough amount of overpayments. He was asked for and made a repayment that accounted for the overpayments plus interest.
"I just said, 'give me the total figure and I'll pay it off immediately'."
Peters would not disclose that amount, saying it was private.
"That is between me and the MSD. It is a private matter, just like the whole leak [to media] came out of IRD, which is a crime. That's the reality of it all."
Asked if he would follow that up with a complaint, Peters said he was too busy to worry about that but believed he was narrowing in on who leaked his information.
Peters said he did not consider disclosing the overpayments to the public when he learnt of the mistake.
"We couldn't identify how it was made. And therefore the only thing was, 'well, how can we fix it up?' Which we did."
