  Using a sample of 225.3 million $\jpsi$ events collected with the BESIII
detector at the BEPCII $e^+e^-$ collider in 2009, searches for the decays of
$\eta$ and $\eta^\prime\to\pi^+ e^- \bar{\nu}_e +c.c.$ in $\jpsi \to \phi \eta$
and $\phi\eta^\prime$ are performed. The $\phi$ signals, which are
reconstructed in $K^+K^-$ final states, are used to tag $\eta$ and
$\eta^\prime$ semileptonic decays. No signals are observed for either $\eta$ or
$\eta^\prime$, and upper limits at the 90% confidence level are determined to
be $7.3\times 10^{-4}$ and $5.0\times 10^{-4}$ for the ratios $\frac{{\mathcal
B}(\eta\to \pi^+ e^- \bar{\nu}_e +c.c.)}{{\mathcal B}(\eta \to \pip\pim\piz)}$
and $\frac{{\mathcal B}(\eta^\prime\to \pi^+ e^-\bar{\nu}_e +c.c.)}{{\mathcal
B}(\eta^\prime \to \pip\pim\eta)}$, respectively. These are the first upper
limit values determined for $\eta$ and $\eta^\prime$ semileptonic weak decays.
