But Rev Dr Brendan MCCarthy, the CoE's national adviser on medical ethics, said there was "anecdotal" evidence that some women were told there was "bad news" if their unborn children were diagnosed with Down's Syndrome - and immediately offered terminations.
Actress and screenwriter Sally Phillips, whose son Olly has Down's, addressed this in the BBC documentary, A world without Down's syndrome.
"The doctor said to us: 'I'm sorry, I'm so sorry.' The nurse on duty cried. I don't think anyone said anything at all positive," she said. "It wouldn't have been any different if they'd told me my child wasn't going to make it."
Dr McCarthy said information given to mothers should include advice on whether they want to undergo NIPT at all.
The NHS test is optional and women who find out their baby has Down's will get support from health professionals to help make a decision on whether to continue their pregancy, NHS England says.
Church of England policy opposes abortion, though recognises that there can be circumstances in which a termination is "the lesser moral difficulty".
"There is a problem that if women are not given up to date information with regard to Down's syndrome, it is possible that it may seem the natural thing to do, the normal thing to do is to go along a pathway that would lead to increased terminations of pregnancy," Dr McCarthy said.
The National Screening Committee has said while it is aware there may be an impact on abortion, the voluntary test was there to give women information and it was up to them what happened next.
NIPT is also offered by private clinics. The Church said these providers should be subject to increased regulation.