The law change has the backing of Labour, the Maori Party, and Act.
It will pass its first reading if New Zealand First support it. The party is generally supportive of medicinal cannabis but will not take a position on the bill until after its next caucus meeting on June 20.
Genter said she felt "incredibly lucky and happy" to have the bill drawn.
"I think it is well past time that New Zealand's outdated drug laws get up to date.
"There are chronically and terminally ill people who are suffering and could get relief, but they are made to be criminals."
Labour's health spokesman David Clark said Labour would back the legislation to select committee, given the aim was to make medicinal cannabis available to those who need it.
"Damien O'Connor has a bill in the ballot for this purpose, and it is our policy to do this very smartly if we are in a position to form a Government in September."
Deputy Prime Minister Paula Bennett said the bill would be discussed in caucus. She was unsure if it would be a conscience vote for National MPs.
United Future leader Peter Dunne said he would disclose his position when the vote is taken.
Also Associate Health Minister, Dunne last week announced restrictions would be removed for cannabidiol, or CBD, bringing New Zealand into line with other countries including Australia.
Until now, CBD has been classed as a controlled drug, meaning the Ministry of Health needed to approve its use. Extracted from cannabis, it has little or no psychoactive properties and can be used for pain relief.
Genter said today that change didn't go far enough, and didn't guarantee medicinal cannabis products would be affordable.
The rule change comes after several high-profile campaigns.
Unionist Helen Kelly took cannabis for pain relief before her death from cancer last year, and urged the Government to allow her to do so legally.
Nelson teenager Alex Renton was the first person to get approval to use CBD before his death in 2015. His mother Rose Renton now has a petition before Parliament which aims to improve access to medical cannabis in New Zealand, at an affordable price.
In February, Dunne agreed to delegate approval for medical marijuana applications to the ministry. Until then, any applications had required ministerial sign-off.