But the memo said that Treasury did not recommend the scheme.
"The Treasury's experience of previous Budgets shows that adding additional analytical requirements to the Budget process likely has little impact on the quality of the bids."
It went on to say that there was a risk that Government departments may treat a gender lens as a "compliance exercise".
University of Auckland Professor Jennifer Curtin – an academic who has done extensive work on gender politics – said the Treasury should "without hesitation" reconsider its position so the process is ready by Budget 2021.
"It is both doable and desirable and could still happen."
Asked about her request being turned down, Genter said she would: "continue to work collaboratively toward solutions, so that all New Zealanders can enjoy the benefits of gender equity".
She added that she "certainly feel the gravity, as Minister for Women, to advocate in government for gender equality".
In a letter sent from Robertson to Genter – also obtained under the OIA – the Finance Minister lays out his reasoning for not setting up the programme.
He said that the Government was committed to gender equality and this was supported by the Treasury's living standards framework, which already has a similar sort of gender lens.
Curtin has been involved in advising the Treasury on gender issues.
She said the proposal, as outlined by the OIA, would have been a way to help track the way policy proposals or programmes might differently impact diverse groups of women and men.
Curtin added that it would have ensured more inclusive, equitable, and sustainable effects of how the Government spends its money.
"It can also help to increase equality, reduce poverty, and address a range of other challenges faced by different groups of women and men."
The Treasury memo was written in August last year, well before the shape and nature of the budget was shifted dramatically due to Covid-19.