The Prime Minister's offer came during extended discussions with Indian counterpart Narendra Modi, who has been putting pressure on May to open the door for more Indian students and skilled workers.
Following talks between the two leaders at Hyderabad House in New Delhi, Mrs May made clear that any further liberalisation of the visa regime must be linked to Indian readiness to take back nationals who overstay their permission to remain in Britain.
As part of a new "strategic dialogue" on home affairs issues, the UK "will consider further improvements to our visa offer if at the same time we can step up the speed and volume of returns of Indians with no right to remain in the UK", said Mrs May.
"And the UK will continue to welcome the brightest and best Indian students, with the latest figures showing that nine out of 10 applications are granted."
UK officials were delighted that Mr Modi granted Mrs May an unusually lengthy 90-minute one-on-one private discussion in talks which formed the centrepiece of her three-day trade mission to the country.
May and Modi also agreed to co-operate on a new cyber security framework, as well as tackling the use of the internet for radicalisation and recruitment attempts by violent extremists.
Speaking alongside Mrs May at a tech sector summit ahead of their working lunch, Mr Modi made clear his concern about visa changes which have almost halved numbers of Indian students at UK universities over the past five years.
"Education is vital for our students and will define our engagement in a shared future," he said.
"We must therefore encourage greater mobility and participation of young people in education and research opportunities."
Following the lunch, he signalled he was also concerned about restrictions on visas for skilled workers.
Welcoming the establishment of a UK-India joint working group on trade, he said: "We believe that such a group should not only focus on trade in goods but also the expansion of services trade, including greater mobility of skilled professionals."
May has already announced moves to speed the way for Indian executives through UK airports as well as helping a handful of top business figures with visa applications.
Aides stressed that the improvements to the visa system on offer did not include any relaxation of eligibility criteria. And no estimates are being made of the number of extra visitors that might result.
Instead, the UK is ready to consider changes such as cutting the cost of visas, speeding up the process or increasing the number of locations where applications can be submitted.
As there is no cap on overall visa numbers, it is possible that such changes may drive up the number of applications made and approved.
Indian nationals are among the most frequent visa overstayers in the UK, and it is thought that thousands may be covered by a deal of the kind being considered.