We present here a thorough photometric analysis of double-barred galaxies, consisting of (i) two-dimensional photometric decompositions including a bulge, inner bar, outer bar, and (truncated) disc; and (ii) three-dimensional statistical deprojections to derive the intrinsic shape of bulges, inner bars, and outer bars. This is the first time the combination of both techniques is applied to a sample of double-barred galaxies. It represents a step forward with respect to previous works, which are based on properties of the integrated light through ellipse fitting and unsharp masking. In this first paper of a series of two, we analyse the nature of the dominant bulges within double-barred systems by using several photometric diagnostics, namely Sérsic index, Kormendy relation, colours, and the better suited intrinsic flattening. Our results indicate that almost all bulges in our sample are classical, whereas only two out of the 17 galaxies under study appear as potential candidates to host secularly formed disc-like bulges. Such result poses the possibility that having a central hot structure may be a requirement for inner bar formation.