  We present the QCD predictions for the azimuthal $\cos 2\varphi$ asymmetry in
charm leptoproduction for the kinematics of the COMPASS experiment at CERN. The
asymmetry is predicted to be large, about 15%. The radiative corrections to the
QCD predictions for the $\cos 2\varphi$ distribution are estimated to be small,
less than 10%. Our calculations show that the azimuthal asymmetry in charm
production is well defined in pQCD: it is stable both perturbatively and
parametrically, and practically insensitive to theoretical uncertainties in the
input parameters. We analyze the nonperturbative contributions to the $\cos
2\varphi$ distribution due to the gluon transverse motion in the target and the
$c$-quark fragmentation. Because of the $c$-quark low mass, the nonperturbative
contributions are expected to be sizable, about (30--40)%. We conclude that
extraction of the azimuthal asymmetries from available COMPASS data will
provide valuable information about the transverse momentum dependent
distribution of the gluon in the proton and the $c$-quark hadronization
mechanism. Finally, we discuss the $\cos 2\varphi$ asymmetry as a probe of the
gluonic analogue of the Boer-Mulders function, $h_{1}^{\perp g}$, describing
the linear polarization of gluons inside unpolarized proton.
