  We analyze the orbital magnetic susceptibility from the band structure of
twisted bilayer graphene. Close to charge neutrality, the out-of-plane
susceptibility inherits the strong diamagnetic response from graphene.
Increasing the doping, a crossover from diamagnetism to paramagnetism is
obtained and a logarithmic divergence develops at the van Hove singularity of
the Moir\'e lattice in the first band. The enhanced paramagnetism at the van
Hove singularity is stronger for relatively large angle but gets suppressed by
the flat spectrum towards the vicinity of the first magic angle. A diverging
paramagnetic susceptibility indicates an instability towards orbital
ferromagnetism with an orbital out-of-plane magnetization and a Landau level
structure. The region of instability is however found to be practically very
small, parametrically suppressed by the ratio of the electron velocity to the
speed of light. We also discuss the in-plane orbital susceptibility at charge
neutrality where we find a paramagnetic response and a logarithmic divergence
at the magic angle. The paramagnetic response is associated with negative
counterflow current in the two layers and does not admit a semiclassical
description.
