Wall's Marriage Equality Bill passed its first reading last year by 80 votes to 40 and it has been sent to a select committee.
The committee will report to Parliament on February 28, and a second reading is scheduled for March 20. A third and final reading could happen in May.
Differences in attitudes toward same-sex marriage were evident in the Herald poll.
The majority of younger people gave it their support and the majority of those 75-plus opposed it.
Wall said she expected her bill would be one of the big issues to be debated at gatherings this holiday season.
"Older people remember when homosexuality was illegal so it is not surprising that it is younger people who are more accepting. Some people, even if they do not agree with it, still don't want to stop someone else's human right to marry."
Females were more supportive of same-sex marriage - more than 60 per cent of women supported legalisation, compared to 47 per cent of men.
New Zealanders interviewed also showed support for voluntary, doctor-assisted euthanasia. More than 60 per cent believed terminally ill people should be able to chose when to end their life.
Visiting voluntary euthanasia expert Yvonne Shaw said she was not suprised at the results. She said New Zealanders were "independent people who would want control at the end of their own lives".
Shaw is from Oregon in the United States, where doctor-assisted suicide is legal. She is in New Zealand to talk about how the same system could work here.
In Switzerland, Oregon, Montana and Washington, doctors can prescribe a lethal dose but the patient alone must administer it.
Voluntary euthanasia, where the patient consents to euthanasia, is legal in Belgium, Luxembourg and the Netherlands.
The Voluntary Euthanasia Society of New Zealand has 1250 members.
Labour MP Maryan Street - who is the author of a member's bill that would legalise assisted suicide in certain cases - decided to act after a meeting with society members a year ago. Despite similar proposals being voted down in 1995 and 2003, she believed the mood of the nation had changed.
POLL
Do you think that same-sex civil unions should be extended to marriage?
• Yes 53.9%
• No 38.1%
• Unsure 8%
Do you think doctors should be allowed by law to end an incurable patient's life, if the patient requests it?
• Yes 60.5%
• No 18.2%
• Depends on situation 16.2%
• Unsure 5.1%
Poll: Key Research. Commissioned by: Herald on Sunday