The demands are contradictory for European leaders, who have emphasised that access to the single market is dependent on allowing free movement of workers.
"I want to give British companies the maximum freedom to trade with and operate within the single market and let European businesses do the same thing here," May told the Conservative party conference.
But she added: "We are not leaving the European Union only to give up control of immigration all over again and we're not leaving only to return to the jurisdiction of the European Court of Justice".
The announcement sent the pound plunging against the euro and dollar but stocks have soared since a weaker pound has helped boost British exporters.
Also Read
"It's going to be a tough negotiation. It will require some give and take," May said today.
However, Downing Street stressed the UK remains open to
the "brightest and the best" from India and that Indians had been issued more work-related visas than the US, China and Australia combined.
"This visit is about highlighting that the UK-India relationship really matters. The UK is the largest G20 investor in India and India is the second largest job creator in the UK. There are strong bonds and ties there and it is important to keep up the momentum," the spokesperson said.
May will be accompanied by international trade secretary Liam Fox and minister of state in the Department for International Trade, Greg Hands.