  Based on 22.7 fb$^{-1}$ of $e^+e^-$ annihilation data collected at 33
different center-of-mass energies between 3.7730 GeV and 4.7008 GeV with the
BESIII detector at the BEPCII collider, Born cross sections of the two
processes $e^+e^-\to \phi K^+ K^-$ and $e^+ e^- \to \phi K_{S}^{0} K_{S}^{0}$
are measured for the first time. No indication of resonant production through
an intermediate vector state $V$ is observed, and the upper limits on the
product of the electronic width $\Gamma_{e^+e^-}$ and the branching fraction
$Br(V\rightarrow \phi K \bar{K})$ of the processes $e^+e^- \to V \to \phi K^+
K^-$ and $e^+e^- \to V \to \phi K_S^0K_S^0$ at the $90\%$ confidence level are
obtained for a large parameter space in resonance masses and widths. For the
current world average mass and width of the $\psi(4230)$ of $m=4.2187$
GeV$/c^2$ and $\Gamma=44$ MeV, we set upper limits on the $\phi K^+ K^-$ and
$\phi K_S^0K_S^0$ final states of 1.75 eV and 0.47 eV at the $90\%$ confidence
level, respectively.
