  We study the importance of interband effects on the orbital susceptibility of
three bands $\alpha$-${\cal T}_3$ tight-binding models. The particularity of
these models is that the coupling between the three energy bands (which is
encoded in the wavefunctions properties) can be tuned (by a parameter $\alpha$)
without any modification of the energy spectrum. Using the gauge-invariant
perturbative formalism that we have recently developped, we obtain a generic
formula of the orbital susceptibility of $\alpha$-${\cal T}_3$ tight-binding
models. Considering then three characteristic examples that exhibit either
Dirac, semi-Dirac or quadratic band touching, we show that by varying the
parameter $\alpha$ and thus the wavefunctions interband couplings, it is
possible to drive a transition from a diamagnetic to a paramagnetic peak of the
orbital susceptibility at the band touching. In the presence of a gap
separating the dispersive bands, we show that the susceptibility inside the gap
exhibits a similar dia to paramagnetic transition.
