  The scalars $f_0$ closest to 1.5 GeV contain the mesons $f_0(1370)$,
$f_0(1500)$ and $f_0(1710)$, and the latter two ones are usually viewed as the
potential candidates for the scalar glueballs. In this work, by including the
important contributions from the vertex corrections, we study the decays $B^0
\to J/\psi f_0$ within the improved perturbative QCD approach and analyze the
possible scalar glueball hunting. Together with the two mixing models, namely,
$f_0(1500) (f_0(1710))$ being the primary scalar glueball in model I (II), and
two classification scenarios, namely, $f_0$ being the $q\bar q$ excited
(ground) states in scenario 1 (2), the branching fractions associated with
their ratios for $B^0 \to J/\psi f_0$ are evaluated comprehensively. The
predictions with still large uncertainties in the considered two mixing models
are roughly consistent with currently limited data, which indicates that both
more rich data and more precise predictions are urgently demanded to figure out
the scalar glueball clearly in the future. Moreover, several interesting ratios
between the branching fractions of $B^0 \to J/\psi f_0(\to \pi^+ \pi^-/K^+
K^-)$ and $B^0 \to J/\psi \rho^0/\phi (\to \pi^+ \pi^-/ K^+ K^-)$ that could
help us to understand the nature of scalar $f_0$ are defined and predicted
theoretically. These ratios should be examined in future experiments.
