  Some models of leptogenesis involve a quasi-degenerate pair of heavy
neutrinos $N_{1,2}$ whose masses can be small, $O({\rm GeV})$. Such neutrinos
can contribute to the rare lepton-number-violating (LNV) decay $W^\pm \to
\ell_1^\pm \ell_2^\pm (q'{\bar q})^\mp$. If both $N_1$ and $N_2$ contribute,
there can be a CP-violating rate difference between the LNV decay of a $W^-$
and its CP-conjugate decay. In this paper, we examine the prospects for
measuring such a CP asymmetry $A_{\rm CP}$ at the LHC. We assume a value for
the heavy-light neutrino mixing parameter $|B_{\ell N}|^2 = 10^{-5}$, which is
allowed by the present experimental constraints, and consider $5~{\rm GeV} \le
M_N \le 80~{\rm GeV}$. We consider three versions of the LHC -- HL-LHC, HE-LHC,
FCC-hh -- and show that small values of the CP asymmetry can be measured at
$3\sigma$, in the range $1\% \lesssim A_{\rm CP} \lesssim 15\%$.
