Four of Mr English's children were born at the centre, which is in ongoing negotiations with the Southern DBH over funding.
In a statement to the Otago Daily Times, Mr English said while DHBs needed to be ''mindful of the impacts of their decisions on local communities and ensure they can adequately meet the needs of rural communities'', the negotiations were between the DHB and Northern Southland Health, and were ''independent".
A director of the company behind the centre believes it can continue to operate until next September, but unless a solution is found to address its funding woes its future after that is uncertain.
Northern Southland Health Ltd director Robyn Crooks said the company, which runs the maternity centre, had not had an increase in its contract fund since 2012.
The Southern DHB's latest offer of a 1% increase was ''pretty unsatisfactory''.
Mrs Crooks said Lumsden was geographically isolated and did not have a pool of midwives.
The sole local midwife was a lead maternity carer (LMC), meaning she was self-employed and travelled throughout the area to deliver babies.
However, because Lumsden was a primary birthing facility, it needed to have a midwife available ''24/7''.
''We actually have to pay somebody to come and stay on site 24/7, just because we don't have anybody within a reasonable area''.
The company was looking at how it could become ''more flexible'', which make see a midwife available within a reasonable area but not necessarily on-site.
But even if that cost was removed ''we would still be running at a deficit'', Mrs Crooks said.
The company believed a 40% increase of its funding would make its operation more viable, particularly because its nursing staff were also underpaid.
''As a medical trust, we are pretty confident that it will stay open in the short term ... [but] if we cannot get further funding, we cannot continue to run at losses.
''We all want to keep Lumsden open, but it's also got to be a viable option - we are not prepared to sell cheese rolls to keep Lumsden open.''
Last Saturday, Te Anau resident Jennifer Haslam organised a march in the township of around 2000 residents. It was attended by more than 200 people, which reflected the community's concern about the potential closure of the facility.
Te Anau's population was growing, with many young couples and families, yet it had no resident midwife, Ms Haslam said.
''The closest midwife is 80km away, and she's fully booked.''
She and her newly-born son received postnatal care at the centre last October after she gave birth at Southland Hospital.
Continuing under-funding by the SDHB could see it go the same way as Tuatapere Maternity Hospital, which closed last year.
Clutha-Southland MP Todd Barclay said in a Facebook post the Lumsden centre was ''not at risk of closure'' and believed ''quite a lot of misinformation'' was being circulated.
''It's disappointing that some people are trying to scaremonger using this threat [of closure], as that does not provide any certainty or comfort to the community.
''What is clear is that the current cost structure of the trust is far too high and unsustainable.
''I will not let this centre close and I have made this very clear.''
In a statement yesterday, DHB chief executive Chris Fleming said the board was working with Northern Southland Health to find the best long-term solution for the community.
-Additionally reported by Guy Williams
