  We investigate the maximum value of the spin-independent cross section
($\sigma_\text{SI}$) in a dark matter (DM) model called the two-Higgs doublet
model + a (THDM+a). This model can explain the measured value of the DM energy
density by the freeze-out mechanism. Also, $\sigma_\text{SI}$ is suppressed by
the momentum transfer at the tree level, and loop diagrams give the leading
contribution to it. The model prediction of $\sigma_\text{SI}$ highly depends
on values of $c_1$ and $c_2$ that are the quartic couplings between the gauge
singlet CP-odd state ($a_0$) and Higgs doublet fields ($H_1$ and $H_2$), $c_1
a_0^2 H_1^\dagger H_1$ and $c_2 a_0^2 H_2^\dagger H_2$. We discuss the upper
and lower bounds on $c_1$ and $c_2$ by studying the stability of the
electroweak vacuum, the condition for the potential bounded from the below, and
the perturbative unitarity. We find that the condition for the stability of the
electroweak vacuum gives upper bounds on $c_1$ and $c_2$. The condition for the
potential to be bounded from below gives lower bounds on $c_1$ and $c_2$. It
also constrains the mixing angle between the two CP-odd states. The
perturbative unitarity bound gives the upper bound on the Yukawa coupling
between the dark matter and $a_0$ and the quartic coupling of $a_0$. Under
these theoretical constraints, we find that the maximum value of the
$\sigma_\text{SI}$ is $\sim 5\times 10^{-47}$ cm$^2$ for $m_A = $ 600 GeV, and
the LZ and XENONnT experiments can see the DM signal predicted in this model
near future.
