She said the businesses should be at a growth phase with series A funding or above already in place.
"They should have a proven solution in the market already that they are hoping to scale into other markets and about nine to 12 months runway, and sufficient resources available for product concept testing with Visa."
Lim said the types of companies it was looking for would fit into four themes.
"With this extraordinary year that we have had we scanned the region and identified four of the most pressing financial and technological opportunities we want to work on together."
Firstly it was building a network of networks - developing new ways of moving money beyond the traditional credit and debit cards.
"It's start-ups connecting to real-time payments, account to account, potentially even digital currencies."
The second theme was empowering small- and medium-sized companies.
"It has been a challenging year for small and medium businesses out there so we are really looking at new ways to support these businesses - how they can manage money, go through digital transformation faster, how they can more easily accept payments."
The third area was around leveraging open data with start-ups looking at different topics like digital identities, what they can do to help make personalised shopping or financial experiences even better and data security, protection and consent.
And the final theme is enabling digital commerce.
"We know there are a lot of consumers and businesses that may be still cash dependent - not so much in New Zealand but some of the growing economies like Indonesia, Vietnam - how can we expand access to them - digital and financial inclusion and really move more users into digital payments."
Businesses have until January 22 to apply through an online form on Visa's website. Those reaching the final stage of the selection process are expected to make a live video pitch.
Lim said those who got through to the programme would gain access to Visa's networks, partners and client stakeholders including banks, merchants, acquirers, and digital wallet operators.
"So it's really opening the doors to that network of partners to help them to scale."
In addition to access and mentorship, Visa would also put its technical expertise behind the startups, helping them through the commercialisation phase.
Those accepted into the programme will be announced in February and Visa will start to work with the businesses from March for four to six months.
While New Zealand was a small part of the Asia-Pacific, Lim said the country was known for its entrepreneurship and growing technology community.
"There are lots of interesting startups that we have our eye on in NZ - we know some of them already through existing relationships and through hero success stories like Xero."
Visa may look to take in an equity stake in the businesses although Lim said that was not guaranteed.