Several subspace algorithms for the identification of bilinear systems have been proposed recently. A key practical problem with all of these is the very large size of the data-based matrices which must be constructed in order to ‘linearise’ the problem and allow parameter estimation essentially by regression. Another shortcoming of currently known subspace algorithms for bilinear systems is that the results are biased for most input signals. This paper focuses on the cause of this bias. A conceptual algorithm which can achieve unbiased estimation under less restrictive assumptions on the system and input signals is presented. It is pointed out that one combination of an existing algorithm and particular conditions on the input is an instance of this conceptual algorithm. Also, the conceptual algorithm may shed light on the trade-off between accuracy and computational complexity which has been noted in our earlier work.