Recently, evidence for a conducting surface state (CSS) below 19 K was reported for the correlated
d
-electron small gap semiconductor FeSi. In the work reported herein, the CSS and the bulk phase of FeSi were probed via electrical resistivity ρ measurements as a function of temperature
T
, magnetic field
B
to 60 T, and pressure
P
to 7.6 GPa, and by means of a magnetic field-modulated microwave spectroscopy (MFMMS) technique. The properties of FeSi were also compared with those of the Kondo insulator SmB
6
to address the question of whether FeSi is a
d
-electron analogue of an
f
-electron Kondo insulator and, in addition, a “topological Kondo insulator” (TKI). The overall behavior of the magnetoresistance of FeSi at temperatures above and below the onset temperature
T
S
= 19 K of the CSS is similar to that of SmB
6
. The two energy gaps, inferred from the ρ(
T
) data in the semiconducting regime, increase with pressure up to about 7 GPa, followed by a drop which coincides with a sharp suppression of
T
S
. Several studies of ρ(
T
) under pressure on SmB
6
reveal behavior similar to that of FeSi in which the two energy gaps vanish at a critical pressure near the pressure at which
T
S
vanishes, although the energy gaps in SmB
6
initially decrease with pressure, whereas in FeSi they increase with pressure. The MFMMS measurements showed a sharp feature at
T
S
≈ 19 K for FeSi, which could be due to ferromagnetic ordering of the CSS. However, no such feature was observed at
T
S
≈ 4.5 K for SmB
6
.