  We investigate the existence of weakly coupled gas-like states comprised of
three $\alpha$ particles around an $^{16}$O core in $^{28}$Si. We calculate the
excited states in $^{28}$Si using the multi-configuration mixing method based
on the $^{16}$O + 3$\alpha$ cluster model. We also include the $^{16}$O +
$^{12}$C and $^{24}$Mg + $\alpha$ basis wave functions prepared by the
generator coordinate method. To identify the gas-like states, we calculate the
isoscalar monopole transition strengths and the overlap of the obtained states
with the geometrical cluster wave function and the
Tohsaki-Horiuchi-Schuck-R\"{o}pke (THSR) wave function. The results show that
the obtained fourth and twelfth states significantly overlap with the THSR wave
function. These two states clearly coexist with the $^{16}$O + $^{12}$C cluster
states, emerging at similar energies. The calculated isoscalar monopole
strengths between those two states are significantly large, indicating that the
states are members of the excitation mode. Furthermore, the calculated
root-mean-squared (RMS) radii for these states also suggest that a layer of
gas-like three $\alpha$ particles could exist around the surface of the
$^{16}$O core, which can be described as a "two-dimensional gas" in the
intermediate state before the Hoyle-like three $\alpha$ states emerge.
