  Inclusive semileptonic $B\to X_c \ell \bar{\nu}_\ell$ decays, where $\ell =
\mu,e$, are by now standard candles in the determination of the CKM element
$|V_{cb}|$. These determinations rely on the heavy-quark expansion and use
moments of decay spectra to extract the non-perturbative parameters directly
from data under the standard model assumption. At the same time, new physics
could influence the moments of the inclusive decay. In this paper, we compute
power-corrections and next-to-leading order corrections in the strong coupling
constant using the full basis of dimension-six new physics operators for the
inclusive $B \to X_c \ell \bar{\nu}$ decay. We provide predictions for lepton
energy, hadronic and leptonic invariant mass moments, and perform a
phenomenological study to show the possible impact of new physics. Our results
could be used to perform a global fit including new physics contributions.
