  We develop a gauge-independent perturbation theory for the grand potential of
itinerant electrons in two-dimensional tight-binding models in the presence of
a perpendicular magnetic field. At first order in the field, we recover the
result of the so-called {\it modern theory of orbital magnetization} and, at
second order, deduce a new general formula for the orbital susceptibility. In
the special case of two coupled bands, we relate the susceptibility to
geometrical quantities such as the Berry curvature. Our results are applied to
several two-band -- either gapless or gapped -- systems. We point out some
surprising features in the orbital susceptibility -- such as in-gap
diamagnetism or parabolic band edge paramagnetism -- coming from interband
coupling. From that we draw general conclusions on the orbital magnetism of
itinerant electrons in multi-band tight-binding models.
