  Inflationary models derived from $f(R)$ gravity, where the scalaron rolls
down with a constant rate from the top to the minimum of the effective
potential, are considered. Specifically, we take into account three $f(R)$
models \textit{i.e.}\,Starobinsky $R^{2}$, $R^{2p}$ and the logarithmic
corrected models. We compare the inflationary parameters derived from the
models with the observational data of CMB anisotropies \textit{i.e.} the Planck
and Keck/array datasets in order to find observational constraints on the
parameters space. We find that although our $f(R)$ constant-roll models for
$\gamma=0$ show observationally acceptable values of $r$, they do not predict
favoured values of the spectral index. In particular, we have $n_{s}>1$ for the
Starobinsky $R^{2}$ and $R^{2p}$ models and $0.996<n_{s}<0.999$ for logarithmic
model. Finally, we study the models from the point of view of Weak Gravity
Conjecture adopting the swampland criteria.
