  Nucleon-nucleon interactions, both bare and effective, play an important role
in our understanding of the non-perturbative strong interaction, as well as
nuclear structure and reactions. In recent years, tremendous efforts have been
seen in the lattice QCD community to derive nucleon-nucleon interactions from
first principles. Because of the daunting computing resources needed, most of
such simulations were still performed with larger than physical light quark
masses. In the present work, employing the recently proposed covariant chiral
effective field theory (ChEFT), we study the light quark mass dependence of the
nucleon-nucleon interaction extracted by the HALQCD group. It is shown that the
pion-full version of the ChEFT can describe the lattice QCD data with
$m_\pi=469$ MeV and their experimental counterpart reasonably well, while the
pion-less version can describe the lattice QCD data with $m_\pi=672, 837, 1015,
1171$ MeV, for both the $^1S_0$ and $^3S_1$-$^3D_1$ channels. The slightly
better description of the single channel than the triplet channel indicates
that higher order studies are necessary for the latter. Our results confirmed
previous studies that the nucleon-nucleon interaction becomes more attractive
for both the singlet and triplet channels as the pion mass decreases towards
its physical value. It is shown that the virtual bound state in the $^1S_0$
channel remains virtual down to the chiral limit, while the deuteron only
appears for a pion mass smaller than about 400 MeV. It seems that proper chiral
extrapolations of nucleon-nucleon interaction are possible for pion masses
smaller than 500 MeV, similar to the mesonic and one-baryon sectors.
