The proposal comes from Jamie Hartzell, whose own social enterprise, the Ethical Property Company, has so far launched two share issues raising more than £5.5m.
Hartzell identified the need for a "positive alternative" to the mainstream markets, but stressed that such an alternative would have to be "carefully crafted" to protect against exploitation for private benefit.
"In the minds of many of the members of the public who make up the ethical investment community, the trading of shares is an unethical practice, leading to profit through speculation with little benefit to society," the report reads.
The idea of a social stock exchange gained momentum in May, when Ed Miliband, former minister for the third sector, chaired a meeting to discuss the possibility of such a scheme.
"I am looking at ideas for an internet virtual cafe for ethical investment, which would not be based on exchange but on information," Hartzell told Third Sector. "One thing that the FTSE provides is a list of eligible companies and details of what they provide."
Separately, Mark Campanale, a director of the UK Social Investment Forum, is working with Pradeep Jethi, a former London Stock Exchange product developer, on a business plan for a social stock exchange.
The pair are looking to raise £200,000 to commission research into how such an exchange could meet Financial Services Authority regulations, how it would function and which technical systems it would use.
"The vision is to enable investors interested in businesses that have a social mission to meet companies and entrepreneurs that have proper business models, and to support investment in companies that have social purposes," Campanale said.
However, not all sector finance experts believe a stock exchange is the best step forward. "Our view is that this may be a bit premature," said Whitni Thomas, investment manager at Triodos Bank.
"At the moment, we should be focusing on helping organisations grow to the size of successful enterprises such as the Ethical Property Company and Cafedirect.
"We don't want to detract time, energy and resources from helping the smaller enterprises to scale up. Only a small percentage of mainstream businesses ever float on the stock exchange. So why are we trying to impose this same model on social enterprises?"
A spokeswoman for the Office of the Third Sector confirmed that ministers were talking to social entrepreneurs and finance experts to determine an archetype for a social stock exchange, but said the department could not comment on any particular initiative.