  Cu$_3$TeO$_6$, a three-dimensional antiferromagnet forming a unique spin-web
lattice of spin-1/2 Cu2+ ions below the Neel temperature T$_N$ = 62 K, has
recently been found to exhibit topological Dirac or nodal magnon dispersion. In
this study, we report the discovery of the linear magnetoelectric (ME) effects
in Cu$_3$TeO$_6$ below TN. Our pyroelectric current measurements at a constant
magnetic field (H) reveal a linear increase of electric polarization (P) with H
for both P // H and P $\perp$ H configurations; a maximum P // [110] = 20
$\mu$C/m$^2$ is obtained at u0H // [1-10] = 14 T, corresponding to a linear ME
coefficient 1.8 ps/m. Magnetic point group analysis and Monte-Carlo simulations
confirm that finite linear ME coefficients are allowed in the off-diagonal and
diagonal ME tensor components, consistent with the magnetic point group of
$\bar{3}'$. As the parity-time symmetry can be broken in the presence of H or
electric field E in the linear ME materials, we envisage that Cu$_3$TeO$_6$
should exhibit a H- or E-induced transformation in the topological magnon
dispersion from a Dirac point/nodal line type into two Weyl point types.
