Theresa May’s hopes of a post-Brexit trade deal with India suffered a hammer blow from Delhi as the prime minister prepared to make her first official visit to the country.

A spokesman for India’s minister of external affairs suggested that a policy brought in by May as home secretary restricting the right of Indian students to stay in the UK after graduation could prove to be a block on any progress.

Before her first bilateral visit outside Europe on Sunday, taking 33 business representatives to India, May said she wished to “reboot” the relationship between the two countries. “The UK and India are natural partners – the world’s oldest democracy and the world’s largest democracy – and together I believe we can achieve great things – delivering jobs and skills, developing new technologies and improving our cities, tackling terrorism and climate change,” she said.

However the Indian government spokesman, Vikas Swarup, told the Observer that May faced tough questions when she arrived in Delhi on immigration and mobility for Indian students and workers in the UK.

“Indian students and people-to-people relations are important pillars of India-UK ties,” he said. “In the last five years or so, the number of Indian students enrolling in UK universities has gone down by almost 50%; from around 40,000 to about 20,000 now. This has happened because of restrictions on post-study stay in the UK.

“We will continue to raise our concerns regarding mobility with the UK. Mobility of people is closely linked to free flow of finance, goods and services.”

The comments came as the former business secretary, Vince Cable, claimed that May’s relations with India would be a major block to making progress on any UK-India deal. He said: “The chances of them getting anything out of India are pretty remote pending a complete volte-face on immigration and visas.”

The former Liberal Democrat MP said that among a series of policy decisions by the Home Office, May’s refusal to allow greater access to the UK for skilled Indian workers, in particular, “screwed up” chances of an EU-India trade deal. He said: “When I was secretary of state I was involved in trying to negotiate the EU-India trade agreement, which didn’t get very far. This myth has been created that we are not able to make progress on deals because of Wallonia or Slovenia. In this particular case, the problem was Britain.

“May in particular was very obstructive of any attempt to make a genuine generous concession, and that was one of the things that screwed up the negotiation.”

In 2010, when she was home secretary, May scrapped the post-study work visa, which allowed foreign students a two-year work permit in the UK after completing a course at a British university. Indian students now find it difficult even to get internships or part-time work while they study in the UK.

In 2013 May tried to introduce a controversial “visa bond” scheme for foreign students from six African and Asian countries, of which India was the biggest, to prevent “high-risk” students staying in the UK after their work permits expired. The scheme, which was not introduced in the end, caused widespread outrage among Indian students at the time and Cable said it was “remembered” by Indian politicians.

May is travelling to India with 30 British businesses, including small- and medium-sized enterprises from every region of the country. A series of commercial deals were expected to be sealed during the visit, delivering 1,370 UK jobs, Downing Street said.

May and the Indian prime minister, Narendra Modi, are due to meet on Monday afternoon for more than two hours of talks on trade, investment, defence and security. Downing Street said May wanted to break down existing trade and investment barriers, and secure agreement to talks that would pave the way for a bilateral trade arrangement once the UK had left the EU. The PM will be accompanied by international trade secretary Liam Fox and trade minister Greg Hands.

Before the visit, May said: “This is a partnership about our shared security and shared prosperity. It is a partnership of potential. I intend to harness that potential, rebooting an age-old relationship in this age of opportunity and with that helping to build a better Britain.”

But Shashi Tharoor, chairman of India’s parliamentary standing committee on external affairs, said May’s anti-immigrant policies were “detrimental” to relations between the UK and India.

He said: “I do believe it is as much in Britain’s interests to attract Indian students as it is in India’s interests for our students to study there.

“The Indian government should ask why this is happening, and whether the withdrawal of the welcome mat sends a broader signal, intended or otherwise, to other Indians.”

“The hope and expectation is that most of the students will return to India with enhanced skills and knowledge, whereas emigrants are lost to India altogether, which gives the Indian government no reason to encourage their departure.

“But again, the broader question is whether these anti-immigrant policies signify an anti-Indian stance which would have broader implications for the relations between the two peoples,” he said.