  Motivated by the measurements of branching fractions of the quasi-two-body
decays $B^0\to K^{*0}(f_0(980)\to )\pi^+\pi^-$ and $B^0\to \rho^0(f_0(980)\to
)\pi^+\pi^-$, we study the charmless $B\to V (f_0(980)\to ) \pi^+\pi^-$ decays
in the perturbative QCD approach. Supposing that $f_0(980)$ is a two-quark
state and mixture of $n\bar n=(u\bar u+d\bar d)/\sqrt 2$ and $s\bar s$ with the
mixing angle $\theta$, we calculate the branching fractions of these decays
with the new introduced $s$-wave $\pi^+\pi^-$-pair wave function. When the
mixing angle $\theta$ lies in the range $[135^{\circ}, 155^{\circ}]$, the
calculated branching fractions of the $B^0\to K^{*0}f_0(980)\to
K^{*0}\pi^+\pi^-$ and $B^0\to \rho^0f_0(980)\to \rho^0\pi^+\pi^-$ decays are in
agreement with the experimental data. The branching fractions of other decays
could be measured in the current LHCb and Belle II experiments. Considering the
isospin symmetry, we also estimate the branching fractions of the
quasi-two-body decays $B\to Vf_0(980)\to V\pi^0\pi^0$, which are half of those
the corresponding decays $B\to Vf_0(980)\to V\pi^+\pi^-$. Moreover, the direct
$CP$ asymmetries of these decays are also calculated, and some of them can be
tested in the current experiments.
