  We present the first search for the leptonic decays $D^{*+}\to e^+\nu_e$ and
$D^{*+}\to \mu^+\nu_\mu$ by analyzing a data sample of electron-positron
collisions recorded with the BESIII detector at center-of-mass energies between
4.178 and 4.226 GeV, corresponding to an integrated luminosity of
6.32~fb$^{-1}$. No significant signal is observed. The upper limits on the
branching fractions for $D^{*+}\to e^+\nu_e$ and $D^{*+}\to \mu^+\nu_\mu$ are
set to be $1.1 \times 10^{-5}$ and $4.3 \times 10^{-6}$ at 90\% confidence
level, respectively.
