  We study the collective modes of gluons in an anisotropic thermal medium in
the presence of a constant background magnetic field using the hard-thermal
loop (HTL) perturbation theory. The momentum space anisotropy of the medium has
been incorporated through the generalized $`$Romatschke-Strickland' form of the
distribution function, whereas, the magnetic modification arising from the
quark loop contribution has been taken into account in the lowest Landau level
approximation. We consider two special cases: (i) a spheroidal anisotropy with
the anisotropy vector orthogonal to the external magnetic field and (ii) an
ellipsoidal anisotropy with two mutually orthogonal vectors describing
aniostropies along and orthogonal to the field direction. The general structure
of the polarization tensor in both cases are equivalent and consists of six
independent basis tensors. We find that the introduction of momentum anisotropy
ingrains azimuthal angular dependence in the thermo-magnetic collective modes.
Our study suggests that the presence of a strong background magnetic field can
significantly reduce the growth rate of the unstable modes which may have
important implications in the equilibration of magnetized quark-gluon plasma.
