A potential multibillion dollar fighter jet deal is also on the table, with officials travelling with Macron expressing confidence that a contract for 114 Rafales can be finalised.
Speaking to reporters after his meeting with Macron in Mumbai, India's Prime Minister Narendra Modi said that "in today's turbulent world, this is a partnership for global stability."
Macron said the "remarkable acceleration" of ties with India came in response to the "changing international order".
Both leaders had earlier referred to each other as their "dear friend", in posts on social media.
Macron is on his fourth visit to India since taking office in 2017, accompanied by his wife Brigitte.
He began his trip in India's financial capital, honouring the victims of the 2008 Mumbai attacks and meeting Bollywood film stars before sitting down with Modi.
The visit follows New Delhi's confirmation last week that it intends to place a major order for Dassault Rafale jets, as well as the signing of a landmark free trade agreement between India and the European Union in January.
Macron will travel to New Delhi for an artificial intelligence summit on Wednesday and Thursday.
New Delhi has sought over the past decade to reduce its dependence on Russia, its traditional main supplier of military equipment, turning to other countries while also pushing for more domestic production.
An Indian defence ministry statement last week said the proposed purchase of Rafale jets had been cleared — with "the majority" of them to be manufactured in India.
Christophe Jaffrelot, an India specialist at Sciences Po Center for International Studies in Paris, described the potential 30-billion-euro ($35-billion) deal for 114 Rafales as the "contract of the century".
If finalised, the jets would add to the 62 Rafales that India has already purchased.
The French presidency has voiced optimism that what it calls a "historic" agreement could be reached.
Modi and Macron also inaugurated via videoconference India's first helicopter final assembly line, a joint venture between India's Tata Group and Airbus.
The facility in Vemagal, in the southern state of Karnataka near the tech hub of Bengaluru, will manufacture the Airbus H125, the company's best-selling single-engine helicopter.
