  In this paper, we study the nuclear medium modifications of the meson
electromagnetic form factors in the framework of the Nambu--Jona-Lasinio (NJL)
model with the help of the Schwinger proper-time regularization scheme to tame
loop divergence and simulate the effect of QCD confinement. In our current
approach, the meson structure and nuclear medium are constructed in the same
NJL model at the quark level. We examine the free space and in-medium charge
radii for the kaon and pion, in addition to the spacelike elastic
electromagnetic form factors of the kaon and pion as well as their quark-sector
form factors, which reflect their internal structure. By comparing to the
experimental data, we found that the free space elastic electromagnetic form
factors for the mesons are consistent with the data, while the in-medium
elastic electromagnetic form factors of the mesons are found to decrease as the
nuclear matter density increases, leading to increase of meson charge radius,
which is consistent with the prediction of other theory calculations. We also
predict the axial nucleon coupling constant $g_A$ in nuclear medium computed
via the Goldberger-Treiman relation (GTR), which is crucial for the search for
the neutrinoless double beta decay ($0\nu \beta \beta$).
