Narendra Modi’s call came as Mrs May announced new schemes to help wealthy Indian businessmen travel to the UK.
Speaking to a technology summit in New Delhi during her three-day visit to the south Asian country, Mrs May said that efforts to bring down barriers to trade must not wait until after Britain left the EU.
And she issued a warning that countries which failed to take advantage of opportunities to trade with like-minded partners will “stagnate” and make their citizens worse off.
Mr Modi welcomed the prospect of enhanced business, scientific and technological collaboration with the UK. But addressing the summit - which took place in a luxury hotel swathed in the thick smog which has forced the introduction of emergency measures in Delhi - he pointedly referred to concerns over visas which have reduced numbers of Indian students at UK universities by almost 50% over the past five years.
Mr Modi said: “Education is vital for our students and will define our engagement in a shared future. We must therefore encourage greater mobility and participation of young people in education and research opportunities.”
Indian officials have blamed visa changes, introduced by Mrs May as home secretary to prevent students from working in Britain after graduation, for a decline from 40,000 to 20,000 in young Indians enrolling in UK universities.
Mr Modi is expected to use talks on Monday to press her for liberalisation of the system.
The Hindustan Times said Britain’s “stringent” visa rules were “casting a shadow” over her visit, quoting an Indian government spokesman as saying: “Mobility issues are of importance to us. We cannot separate free movement of people from free flow of goods, services and investment.
“We hope for UK being responsive to our concerns.”
But as she arrived in India, Mrs May signalled she would resist pressure to relax the regime, insisting the UK already had a “good system” for applications from India.
“The figures show that we issue more work visas to India than (to) the US, Australia and China put together,” said the PM.
“Nine out of 10 visa applications from India are already accepted. So we have, I believe, a good system.”
While Britain is barred from signing bilateral trade deals with third countries until it has left the EU, Mrs May said that there were steps that can be taken immediately to “break down barriers and make it easier to do business”.
Britain and India need to “identify what more we can do now to unleash our businesses, industries, exporters and investors”, she said, adding: “This does not need to wait for us to leave the EU.”
Among initiatives to foster trade, Indian tycoons and their families are to gain access to the Great Club programme which provides assistance with visa-processing, while an estimated 10,000 executives are to benefit from a Registered Traveller Scheme to speed their way past queues at UK airports.
Mrs May declared: “As Britain leaves the EU, we’re determined not to turn our backs on the world but to forge a new, global, outward-looking role for ourselves.
“Because we know from history what happens when countries do not embrace the opportunities of the world. They stagnate. They get poorer. They don’t protect their people; they make them worse off.
“Of course, no country owes any other country a living. But we stand the greatest chance of success when we work with partners with whom we share similar values, legal systems, approaches to business, and ways of looking at the world.”
Mrs May, who has has come to India at the head of a 33-strong UK business delegation, announced a new India-UK Urban Partnership to develop “smart cities” in the sub-continent.
Downing Street estimates the number of new jobs created by Indian investment in the UK in 2015/16 at 7,105, and officials said a further 1,370 are expected to be created by commercial deals due to be sealed during the PM’s three-day trip.