NEW DELHI—U.K. Prime Minister Theresa May on Monday said she wants Britain to be the most passionate advocate of free trade around the world, pushing a vision of globalization at a time that governments and voters have expressed doubts about its benefits.

In India on a two-day visit focused on trade, Mrs. May announced several investments in Indian companies and said the U.K. may make improvements to its visa system for Indians, but didn’t commit to any changes that would result in significant increases of Indians entering the U.K.

The British leader must strike a balance as she pursues new deals in the wake of the U.K. vote to leave the European Union, a decision driven in large part by a desire for more control over immigration and skepticism about trade. Her goal of reducing immigration may get in the way of forming deals quickly, since countries such as India are likely to want concessions on visas.

India under the leadership of Prime Minister Narendra Modi, has tried to modernize and open up its economy, but it has still lagged behind others in the region in lowering commercial barriers. A free-trade deal with the EU has been stalled for years. Moreover, some Indian companies are wary, saying a pact between India and the Association of Southeast Asian Nations triggered a flood of Thai imports.

Ahead of a meeting with Mr. Modi, the U.K. leader argued that countries that don’t embrace opportunities stagnate.

“Free trade creates a rising tide that lifts all boats,” she said. “It makes us all richer. It creates jobs.”

A key sticking point in U.K.-India relations has been Britain’s reluctance to loosen restrictions for Indians wanting to work or study in the U.K. Mrs. May defended the U.K. visa system, saying it worked well and that the U.K. wanted to attract the best of India while keeping migration down.

“The U.K. 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 U.K.,” she said.

Earlier, Mr. Modi briefly raised the issue, saying education will define engagement between India and the U.K. in the future. “We must therefore encourage greater mobility and participation of young people in education and research opportunities,” he said.

The U.K. can’t complete trade deals while still a member of the EU, but is in preliminary discussions on trade with countries including Australia and India. Any deal is likely to take years to complete.

ENLARGE

U.K. and Indian officials said they would meet at a later date to discuss how they would work through challenges, a U.K. official said. The official said Mrs. May and Mr. Modi touched on the visa issue in their meeting earlier Monday.