  We work out the low-energy expansion of the anomalous
$f_1(1285)\to\rho\gamma$ decay amplitude by using the Nambu--Jona-Lasinio model
with $U(2)\times U(2)$ chiral symmetric four-quark interactions in the
one-quark-loop approximation. The related processes $f_1(1285)\to\omega\gamma$,
$a_1(1260)\to\omega\gamma$, and $a_1(1260)\to\rho\gamma$, are also considered.
An effective meson Lagrangian responsible for $f_1\rho\gamma$,
$f_1\omega\gamma$, $a_1\rho\gamma$ and $a_1\omega\gamma$ interactions is found.
The predicted radiative decay widths, $\Gamma_{f_1\to\rho^0\gamma}=311\
\mbox{keV}$, $\Gamma_{f_1\to\omega\gamma}=34.3\ \mbox{keV}$,
$\Gamma_{a_1\to\rho^0\gamma}=26.8\ \mbox{keV}$,
$\Gamma_{a_1\to\omega\gamma}=238\ \mbox{keV}$, allow an experimental test of
the hypothesis that $f_1(1285)$ and $a_1(1260)$-mesons have a quark-antiquark
nature. At present, only the $f_1(1285)\to\rho\gamma$ decay has been measured.
Our result is in remarkably good agreement with the recent data of CLAS
Collaboration $\Gamma_{f_1\to\rho^0\gamma}=453\pm 177\ \mbox{keV}$, but
disagrees with the PDG-based estimate of $\Gamma_{f_1\to\rho^0\gamma}=1326\pm
313\ \mbox{keV}$. The calculations presented require a minimum of theoretical
input, and are shown to be consistent with the non-renormalization theorems of
QCD.
