Supporters say it is not right that Cornwall is the only county in the country without a major venue of this kind.
They want to see a 15,000-seater stadium near Truro, saying that such a centre could provide a home for top-class sport and training facilities in Cornwall as well as a venue for concerts, accommodation and restaurants.
Nearly 800 people have already joined a campaign group on the online social networking site Facebook and a website has been set up to promote the idea.
Peter Marks, chairman of the Stadium for Cornwall group, is hoping for backing from Cornwall's new unitary authority to push the scheme ahead.
He said: "I think this would be a good opportunity for the new authority to unify the whole county with something which is going to be good for everyone in Cornwall.
"All of our county matches could be played there for football and rugby, but it's not just about that – it could be used to host a Celtic games festival, athletics meetings and disabled sporting events."
Mr Marks said the concept was at a very early stage – no specific site has been located and money would have to be found to fund the project.
Mr Marks said: "At the moment the detail has not been decided.
"We would have to decide where the funding comes from but there are lots of different pots such as European money, the sporting associations and the RDA.
""My job is to increase awareness of what we might have in Cornwall if everyone is behind the idea."
Kevin Heaney, the chairman of Truro City FC, has also taken part in detailed discussions about joining forces with the Cornish Pirates rugby club to build a new stadium for Cornwall at Threemilestone, near Truro.
Peter McGahan, managing director of Worldwide Financial Planning, is one of those backing the scheme as he believes the stadium is vital for the continued development of Cornwall.
He said: "Cornwall has undergone so many positive developments over the past few years
"There has been so much success across all fields that to not have a fantastic sporting venue which can feature the best sports people, as well as inspire young people, is just wrong.
"In fact it's even worse when you consider Cornwall is the only county in the country without a major stadium."
Mr McGahan pointed to success stories such as the Eden Project as proof that fantastic things can happen in the county.
"We've proved that you can be the best and be based in Cornwall," he said.
"Cornwall has an abundance of talent who can ensure a stadium happens, as well as success within it, but what we need is more people and businesses to be inspired and to support it."
For more information about the campaign, visit www.stadiumforcornwall.com