While hospital staff were able to access a small amount of donor milk, the limited supply meant only her smallest, Maddilyn, could have it.

That could change, if a motion from Labor backbencher Tara Cheyne passes the ACT Legislative Assembly on Wednesday.

Ms Cheyne will call on the ACT government to investigate setting up a milk bank in Canberra to improve the availability and supply of donor breast milk.

Her motion is understood to have support from Health Minister Meegan Fitzharris, the Liberals' spokeswoman for women Giulia Jones and the ACT Greens.

Ms Cheyne said the Centenary Hospital for Women and Children receives donated milk from an interstate charity, but it's only available for premature babies or if the mothers are unwell and can't express milk.

There are also Facebook groups where mothers can informally donate milk, however the milk is not screened like in a professional milk bank.

"It is obvious there are women in the ACT who have excess breast milk who would like to be able to share their milk with other parents in the ACT community and to do so easily," Ms Cheyne said.

"It's also obvious there are parents who are seeking donated breast milk for a range of reasons - insufficient supply, illness, an inability to suckle. The fact that informal donation and recipient arrangements exist in the ACT demonstrates the need is there - but in those arrangements breast milk is not necessarily screened. A milk bank would provide a central location to donate and to receive, as well as to provide appropriate protections through screening."

Ms Cheyne's motion comes after representations from the Australian Breastfeeding Association about the need for the service in Canberra.

"They explained the benefits of donated breast milk - both to the recipients and the donors - but that physical places to donate, store and distribute breast milk - milk banks - are not common in Australia. After speaking with more and more people in the community about their experiences it became apparent there is a real need for a milk bank in the ACT," Ms Cheyne said.

"Stories which have struck me the most have been from parents who had stillborn children. In some cases, those parents were able to donate their breast milk to another family and, in doing so, have described how they felt were honouring their precious child. But for some parents, they wanted to, for the same reasons, and to assist with their grieving, but were not able to. The establishment of an ACT region milk bank would assist parents across all these circumstances."

Mrs Keen is one of those who would have benefited from a dedicated milk bank.

Staff had to bend the rules to allow her babies to access the donated milk, as they were born one day too late to be officially classified as premature.

"I had emergency surgery and lost a lot of blood. I had a lot of problems expressing milk, it was a really stressful situation. I really wanted to breastfeed, I'd expressed feelings before I had them that was my preference. But I didn't realise what it felt like to be a mother who couldn't provide for their baby until I was in that situation," Mrs Keen said.

Because there was so little donated milk in stock, only her smallest baby, Maddilyn, could have it.

Seven months on, Aleisha, Maddilyn and Eloise are all thriving and Mrs Keen is incredibly thankful that store of milk was there.

"My babies were already struggling so they really needed the best start," Mrs Keen said.