The 48-member Nuclear Suppliers Group (NSG) sets global rules for international trade in nuclear energy technology, and was established in 1974 as a response to India's first nuclear test. New Zealand is a member of the NSG.
India wants in, despite not being a member of the Nuclear Non-Proliferation Treaty, which requires members to make commitments on nuclear disarmament and non-proliferation.
It needs a unanimous vote to join, and has been blocked by Beijing, which has argued signing the non-proliferation treaty is a pillar of the group, and the rules shouldn't be bent for India.
Key has said New Zealand and other "like-minded" countries had said it was possible for India to join without signing the treaty, but that would require them to agree to meet similar conditions.
Joining the NSG is seen as a key test of Modi's foreign policy, and a report by the Economic Times, the world's second-most read English language business newspaper, quoted well-placed sources as saying the "primary objective of hosting Key is to elicit his country's support for NSG membership".
"It is understood that while the top Kiwi political leadership is favourably disposed to India's entry, the New Zealand foreign ministry has been dragging its feet on the matter because of India's non-NPT status," the August report stated. "India is hoping Key's visit would help bridge that gap."
Modi, India's 14th prime minister and a Hindu nationalist, has previously lobbied Key on the nuclear issue, as has US President Barack Obama.
The US position is India needs a sustainable energy alternative to coal power for its 1.3 billion people, and that it is a stable democracy.
Key has rejected the idea that New Zealand softening its stance could be a possible bargaining chip in free-trade negotiations, but said the issue would be raised in his talks with Modi.
"We see them as non-related, but important discussion points in terms of the relationship overall, and it's a good relationship with India and has been growing closer over time."
NZ-India links 'warm but a little bit empty'
"Warm but a little bit empty" - that's the verdict on the India-New Zealand relationship of one senior figure involved in the push to strengthen ties.
Professor Sekhar Bandyopadhyay of Victoria University of Wellington and head of the NZ India Research Institute heard the description while leading a track two delegation to India last year.
People-to-people links were well ahead of the political and economic relationship between the countries, he said.
A common language, shared Commonwealth heritage and Parliamentary democracy, cricket - Brendon McCullum is on the Prime Minister's delegation to India this week - and a growing Indian diaspora in New Zealand delivers the "warm".
But the political relationship hasn't yet matched that. New Zealand has been frustrated in attempts to sign a free trade agreement.
New Zealand made India the subject of its first "NZ Inc" strategy in the same year, with the goal of shipping $2 billion of goods to India by 2015.
That hasn't been met, with goods trade at only $656 million. Negotiations for a FTA kicked off in April 2010 and have gone through 10 rounds without a result.
New Zealand hasn't been alone in struggling to get a trade deal.
Narendra Modi's Government, which swept to power in May 2014, has a "Make in India" doctrine focusing on attracting foreign direct investment rather than free trade.
"For any government in India it is difficult to sell an FTA which proposes to open up the agricultural market. And that's the sticking point, I think. There are lobbies within the ruling party that will oppose this vehemently," Bandyopadhyay said.
However, New Zealand could benefit from India's desire for technology transfer and collaboration in certain sectors, he said, including agriculture technology, cold storage and medical technology.
Four things to know about Narendra Modi
1. Known for his fashion sense, particularly his signature long tunic, the "Modi kurta ". Wore a pinstripe suit during a meeting with US President Barack Obama with the "stripes" actually made up of thousands of letters spelling his name vertically.
2. Put forward a successful resolution to the United Nations for June 21 to become "International Yoga Day". Is a vegetarian known for his work ethic.
3. Anti-Muslim riots broke out during his time as chief minister of the western Indian state of Gujarat. Questions over whether he did enough to stop or discourage them led to travel bans by the US, UK and European Union.
4. The first Indian Prime Minister born after independence from the UK. The pro-business, Hindu nationalist was born to a family of grocers and helped run a tea stall while growing up.
