  We investigate the possibilities of observing the decay mode for $^{124}$Xe
in which two electrons are captured, two neutrinos are emitted, and the final
daughter nucleus is in its ground state, using dark matter experiments with
liquid xenon. The first upper limit of the decay half-life is calculated to be
1.66$\times10^{21}$ years at a 90% confidence level (C.L.) obtained with the
published background data from the XENON100 experiment. Employing a known
background model from the Large Underground Xenon (LUX) experiment, we predict
that the detection of double-electron capture of $^{124}$Xe to the ground state
of $^{124}$Te with LUX will have approximately 115 events, assuming a half-life
of 2.9 $\times$ 10$^{21}$ years. We conclude that measuring $^{124}$Xe 2$\nu$
double-electron capture to the ground state of $^{124}$Te can be performed more
precisely with the proposed LUX-Zeplin (LZ) experiment.
