We study the capture rate in the doublet hyperfine initial state for the muon capture reaction $μ^- + \,^2{\rm H} \rightarrow ν_μ+ n + n$ ($Γ^D$) and the total capture rate for the reaction $ μ^- + \,^3{\rm He} \rightarrow ν_μ+ \,^3{\rm H}$ ($Γ_0$). We investigate whether $Γ^D$ and $Γ_0$ could be sensitive to the $nn$ $S$-wave scattering length ($a_{nn}$). To this aim, we consider nuclear potentials and weak currents derived within $χ$EFT. We employ the N3LO chiral potential with cutoff $Λ$=500 MeV, but the low-energy constant (LEC) determining $a_{nn}$ is varied so as to obtain $a_{nn}$=-18.95 (the present empirical value), -16.0, -22.0, and +18.22 fm. The last value leads to a $nn$ bound state with a binding energy of 139 keV. The LECs $c_D$ and $c_E$, present in the three-nucleon potential and axial-vector current, are fitted to reproduce the $A=3$ binding energies and the triton Gamow-Teller matrix element. The capture rate $Γ^D$ is found to be 399(3) s$^{-1}$ for $a_{nn}$=-18.95 and -16.0 fm; and 400(3) s$^{-1}$ for $a_{nn}$=-22.0 fm. For $a_{nn}$=+18.22 fm, we obtain 275(3) s$^{-1}$ (135(3) s$^{-1}$), when the final $nn$ system is unbound (bound). The rate $Γ_0$ is found to be 1494(15), 1491(16), 1488(18), and 1475(16) s$^{-1}$ for $a_{nn}$=-18.95, -16.0, -22.0, and +18.22 fm, respectively. The theoretical uncertainties are due to the fitting procedure and radiative corrections. Our results seem to exclude the possibility of constraining a negative $a_{nn}$ with an uncertainty of less than $\sim \pm$ 3 fm through an accurate determination of the muon capture rates, but the uncertainty on the present empirical value will not complicate the interpretation of the (forth-coming) experimental results for $Γ^D$. Finally, a comparison with the already available experimental data discourages the possibility of a bound $nn$ state.

5 pages, 2 figures; revisited version accepted for publication on Phys. Rev. C