Hi there, I am not sure about letting agencies or the like, but are your debts unrelated to previous rental properties?? Does the person requesting a reference also run a credit check? This would be the only way that they could discover any debt problems that you have and you would have to give your permission for them to access this info.
Hope this is of some help, but your best source of info would be citizens advice bureau, they have experts who can advise free of charge.