Whitby has offered a £50,000 bursary to the Academy to carry out the work, and provide what he described as "a more balanced approach" to the energy debate.
"Before people start investing in new nuclear power stations we need to know if it really is carbon free," said Whitby.
He claimed that there was no definitive research on the true financial and carbon cost of new nuclear power stations.
"What I feel is important is that this work is carried out by an august body like the Royal Academy and not by some other lobby group."
Whitby made his challenge to the Academy's chief executive Philip Greenish and has a meeting arranged in February to discuss his offer to fund the work.
He has criticised the Academy's 2004 report on the cost of electricity generation for lacking balance. This was carried out by consultant Parsons Brinckerhoff Power.
Greenish denied Whitby had formally offered the Academy money. Accepting money for research depends on several factors, he said. "Our simple interest is in producing evidence-based advice," said Greenish."If this is what he proposes, then it could be helpful," he said.
The Nuclear Industry Association said that an independent report by Parliamentary Office for Science & Technology (POST) showed that nuclear power's carbon footprint was "on a par" with wind power.
It showed that nuclear and wind energy produce 5kg of carbon per MW hour compared to 400kg for gas and 800kg for coal.