But Mr Cameron made clear that in return, he wanted India to tear down outdated barriers to investment to help UK-based companies in areas like insurance and banking to establish a foothold in the fast-growing economy.
Speaking on the first day of his second visit to India as PM, Mr Cameron revealed he was talking to the government in Delhi about the prospects for a new corridor of development between Mumbai and Bangalore, featuring new towns and infrastructure, which could provide opportunities for British planners, architects, construction firms and finance specialists.
Mr Cameron announced the creation of a new "same-day" visa service for Indian businessmen, in response to unhappiness with the current system which can take three days or more to process applications.
And he said there was "no limit" to the number of Indians who would be allowed to study at UK universities and stay on in graduate-level jobs after they qualified.
Speaking to workers at the Mumbai headquarters of the Anglo-Dutch Unilever group, Mr Cameron said: "Britain is one of the most open, easy-to-invest-in countries in the world. We are incredibly welcoming.
"I am very proud of the fact that it is an Indian company, Tata, that makes the Jaguars and Land Rovers that are taking the world by storm. They also roll most of our steel and own Tetley Tea - and you don't get any more British than Tetley Tea.
"I am very proud of that. Britain is an open economy and we encourage that investment.
"I think, in return, we should be having a conversation about opening up the Indian economy, making it easier to do business here, allowing insurance and banking companies to doo more foreign direct investment into the Indian economy.
"There are still many rules and regulations in the Indian economy associated with how you did things in the past which, if you change them, will make your economy grow and deliver more jobs, more wealth, more prosperity across your country.
"It's a good to have, but it goes both ways. We should look at the things we need to do to take our barriers down, and we hope your government will do the same."
Mr Cameron is accompanied on his three-day trip by a delegation of more than 100 representatives of major corporations, small businesses and academic institutions, as well as football's Premier League, the London Underground and nine parliamentarians.
He told his audience in Mumbai that he wants Britain's partnership with India to be "a really special relationship" for the 21st century.
"Britain wants to be your partner of choice," said Mr Cameron. "We think there are huge ties of history and language and culture and business, but we think we have only just started on the sort of partnership we could build."
Britain was "rewriting the rules" in response to Indian business and government calls for easier access to high technology, he said.
And he added: "We are looking with your government at whether we can open up a whole corridor between Mumbai and Bangalore of growing towns and developments and work and plan that with you.
"With me, I have got architects, planners and private finance experts who can work out a complete solution, putting together all the things you need to have that infrastructure."
On providing new infrastructure, Mr Cameron said his main advice was to "start now", rather than delaying construction for years. He cited the example of the London Underground, which deteriorated for decades for lack of investment in upgrades.
"One thing I would say is, if you know you have got to build new infrastructure, even though you know it is going to take many years, start now," he said.
In a round of TV interviews in Mumbai, Mr Cameron rejected suggestions that his desire for a "special relationship" with India was not reciprocated in New Delhi, as Prime Minister Manmohan Singh is yet to visit him in the UK.
Mr Cameron said: "I think the signs are very good. Half of Indian investment into Europe comes to Britain and Britain is now the largest European investor into India.
"I think the basis for that special relationship and partnership is absolutely there.
"This is going to be the third largest economy in the world by 2030 and I want to make sure it is British firms that are helping to build those hospitals, construct those roads, provide those universities, and we want a real exchange between our countries."
Mr Cameron was asked if entrepreneur Lord Bilimoria was right to warn that the UK does not sell itself hard enough in countries like India.
The PM replied: "I think he has a point. I am a great believer that you have got to fill up aeroplanes with business people, get out to these fast-growing countries and sell your country as hard as you can.
"I have now taken trade missions to every G20 country bar one and I am back in India for the second time in two and a half years.
"This is a fast-growing economy, it is going to be one of the stars of the future. We have got to get in there early and that is what British firms are doing."
Mr Cameron was asked whether his appeal to Indian businessmen and students to come to the UK was undermined by the Government's message on getting net annual immigration down into the tens of thousands.
"We want to have a proper policy of controlled immigration and immigration has fallen by a quarter under our Government," he said.
"But we want to make sure that we are attracting at the same time the best and the brightest. We do want to see Indian university students and there is no limit on those who come, if they have got a university place.
"And in terms of our visa operation here in India, it is the biggest one we have anywhere in the world. Nine out of 10 of those who apply for a visa get one.
"We are introducing today a same-day visa service for business people who want to come to Britain for linking up their businesses for trade and other things like that.
"What we had to do was to turn round an immigration system that was completely broken. What we have done is close down 180 bogus colleges that were bringing people in not to study but to work, and often in unskilled jobs.
"We have shut those, but now we can send out the message that (for) our universities, really renowned and seen as high-quality across the world, there is no limit on the numbers who can come to those universities.
"That's good for our universities, it's good for the students who come, and if they work in a graduate job afterwards they are really contributing to our economy.
"It's been turning round a tanker - a totally broken system that we inherited and now one that emphasises quality rather than just quantity."
Mr Cameron later laid a wreath at the memorial to the 16 Indian police officers who died in the terror attacks in Mumbai in 2008.
The Prime Minister bowed his head in respect before the memorial as an honour guard played the Last Post.
Some 164 people were killed in a series of 12 co-ordinated shooting and bombing attacks across Mumbai, which were blamed on terror group Lashkar-e-Taiba.
