The Economic and Social Research Council, which is co-funding the project, recognises only organisations that operate its Collaborative Awards in Science and Engineering accreditation scheme.
Bristol Business School at the University of the West of England and the Centre for Government and Charity Management at London South Bank University will not be submitting bids before the 27 March deadline because neither faculty operates the scheme.
Professor Alex Murdock, head of the centre at London South Bank University, confirmed that his faculty could not bid because it did not hold the relevant qualification. Adrian Sargeant, professor of non-profit marketing at Bristol Business School, said his faculty was in the same situation.
"The ESRC will limit the institutions that could apply and potentially compel applicants to focus on those faculties in their institutions that have the Case accreditation rather than the best voluntary sector expertise," said Sargeant.
Joe Saxton, who co-founded charity research company nfpSynergy, said the sector would lose out.
"The ESRC's funding criteria are nothing to do with the people the research is targeted at," he said.
"It is all to do with its own accreditation system, which is utterly irrelevant and will only narrow the field of candidates."
The centre will conduct research that will underpin policy relevant to the voluntary sector. It is being funded for five years by the ESRC, the Office of the Third Sector and the Barrow Cadbury Trust.
It will consist of a single research centre and three separate institutions known as 'clusters', which will each receive £1m to disseminate research to charities.
A spokeswoman for the ESRC said: "Recognition is a quality assurance mechanism that is put in place to address our key concern that students receive high-quality research training. We cannot be expected to move beyond this framework of quality assurance."
The OTS said the ESRC framework would boost standards.