  Modified gravity theories include $f(\mathbf{R})-$gravity models that are
usually constrained by the cosmological evolutionary scenario. However, it has
been recently shown that they can also be constrained by the signatures of
accretion disk around constant Ricci curvature Kerr-$f(\mathbf{R}_{0})$ stellar
sized black holes. Our aim here is to use another experimental fact, viz., the
terrestrial Sagnac delay to constrain the parameters of specific
$f(\mathbf{R})-$gravity prescriptions. We shall assume that a
Kerr-$f(\mathbf{R}_{0})$ solution asymptotically describes Earth's weak gravity
near its surface. In this spacetime, we shall study oppositely directed light
beams from source/observer moving on non-geodesic and geodesic circular
trajectories and calculate the time gap, when the beams re-unite. We obtain the
\textit{exact }time gap called Sagnac delay in both cases and expand it to show
how the flat space value is corrected by the Ricci curvature, the mass and the
spin of the gravitating source. Under the assumption that the magnitude of
corrections are of the order of residual uncertainties in the delay
measurement, we derive the allowed intervals for Ricci curvature. We conclude
that the terrestrial Sagnac delay can be used to constrain the parameters of
specific $f(\mathbf{R})$ prescriptions. Despite using the weak field gravity
near Earth's surface, it turns out that the model parameter ranges still remain
the same as those obtained from the strong field accretion disk phenomenon.
