  The fundamental resonances of near-extremal Kerr black holes due to massive
scalar perturbations are derived {\it analytically}. We show that there exists
a critical mass parameter, $\mu_c$, below which increasing the mass $\mu$ of
the field increases the oscillation frequency $\Re(\omega)$ of the resonance.
On the other hand, above the critical field mass increasing the mass $\mu$
increases the damping rate $\Im(\omega)$ of the mode. We confirm our analytical
results by numerical computations.
