Tony Tinley responded to The Sunday Times report that the company may spend research budgets overseas if government support for innovation is cut.
Mr Tinley said he wanted the firm to commit itself to the city.
Rolls-Royce said it invests in research and development "where the environment is supportive".
The union fears that grants offered by the department for Business, Innovation and Skills, to firms like Rolls-Royce, could be turned into loans as a result of spending cuts.
Mr Tinley, a senior regional Unite officer in the East Midlands, said: "If you move [research and development], the manufacturing goes with it.
"You could be arguing that the whole presence of Rolls-Royce is being put at risk.
"I get called a scaremonger for this but when I've sat and looked in the eyes of these people [senior management], I don't see that commitment."
Dr Ian Jackson, from Staffordshire University's business school, said turning grants into loans could have a serious effect on Rolls-Royce.
He said: "It would put at risk the qualified scientists and engineers not just at Rolls-Royce in Derby, but also the supply chain, and wider into the hinterland of what supports that."
Rolls-Royce, which employs about 13,000 people in Derby, said in a statement: "Rolls-Royce has always maintained that we invest in research and development where the environment is supportive.
"The location of research and technology is dependent on where the early stages of innovation happens, and the UK today has a compelling and competitive environment for Rolls-Royce to invest."
A government spokesperson said: "We want the UK to be the best place in Europe to innovate and grow a business.
"We are simplifying support for business, boosting collaboration and investing in research infrastructure on a record scale."
Hundreds of jobs could be lost if Rolls-Royce moves it research and development work out of Derby, a Unite union organiser has said.