A draft guidance document called Cohesion Guidance for Funders was published this week by the Department of Communities and Local Government.
It does not endorse a recommendation made by the Commission on Integration and Cohesion - a body set up two years ago to advise government on diversity - that councils should focus on providing funding for organisations that work across the population rather than with single groups.
It says councils should instead be aiming to promote community cohesion through the voluntary sector projects they fund.
Kevin Curley, chief executive of community umbrella body Navca, said his organisation was relieved the guidance did not stipulate that funding for single groups should be the exception rather than the rule.
But Vandna Gohil, director of Voice4Change England, which represents BME organisations, said that BME charities could lose money if they did not push for recognition with councils.
"There is a huge potential for organisations that are not vocal about this to miss out," she said.
"Our experience is that organisations affected by policy are often those that are the furthest from the policy table."
Ann Blackmore, head of policy at umbrella body the NCVO, said the guidance could increase the burdens on those applying for funding.
"This would run counter to guidance from the Treasury and the National Audit Office that government funders should apply controls that are proportionate to both the size of the funding and the risk connected with it," she said.