  The origin of sub-TeV gamma rays detected by the \textit{Fermi}-Large Area
Telescope (LAT) from the \textit{Fermi} Bubbles (FBs) at the Galactic center is
still uncertain. In a hadronic model, acceleration of protons and/or nuclei and
their subsequent interactions with gas in the bubble volume can produce the
observed gamma rays. Recently the High Altitude Water Cherenkov (HAWC)
observatory reported an absence of gamma-ray excess from the Northern FB at
$b\gtrsim 6^\circ$ Galactic latitude, which resulted in flux upper limits in
the energy range of $1.2-126$ TeV. These upper limits are consistent with the
gamma-ray spectrum measured by \textit{Fermi}-LAT at $|b|\ge 10^\circ$, where
an exponential cutoff at energies $\gtrsim 100$ GeV is evident. However, the FB
gamma-ray spectrum at $|b|\le 10^\circ$, without showing any sign of cutoff up
to around 1 TeV in the latest results, remains unconstrained. The upcoming
Cherenkov Telescope Array (CTA) will perform a Galactic center survey with
unprecedented sensitivity in the energy between 20 GeV and 300 TeV. In this
work, we perform both morphological and classic on/off analyses with planned
CTA deep central and extended survey and estimate the sensitivity of CTA to the
FB hadronic gamma-ray flux models that best fit the spectrum at $|b|\le
10^\circ$ and whose counterpart neutrino flux model best fits the optimistic
neutrino spectrum from IceCube Neutrino Observatory. We also perform
sensitivity analysis with a future ground-based Cherenkov detector the Large
High Altitude Air Shower Observatory (LHAASO). We find that CTA will be able to
discover or constrain the FB gamma-ray flux at $|b|\le 10^\circ$ in the
$\approx$ 200 GeV -- 100 TeV range with planned observation strategy, while
LHAASO may constrain emission in the $\approx 100$~GeV -- 100~TeV range if
$\lesssim 10\%$ systematic uncertainties can be achieved.
