  The $J/\psi$-$p$ scattering length, $\alpha_{J/\psi p}$, can be extracted
from the $J/\psi$ photoproduction cross section near threshold using the Vector
Meson Dominance (VMD) model to relate the reaction $\gamma p \to J/\psi p$ to
$J/\psi p \to J/\psi p$. Such estimates based on experimental data result in
values for $|\alpha_{J/\psi p}|$, which are much lower than most of the
theoretical predictions. In this work, we study the relations between the
different results, depending on the use of the total or the differential cross
sections, and the method of extrapolating the data to threshold in the case of
a low-statistics data sample, such as the near threshold $J/\psi$
photoproduction dataset. We estimate a range for $|\alpha_{J/\psi p}|$ of
$0.003$ to $0.025$~fm as extracted from experimental data within the VMD model
and discuss possible reasons for such lower values compared to the theoretical
results.
