  We consider two possible sources of the $T$-violating muon transverse
polarization in the process $K^+ \to \mu^+
  \nu \gamma$: electromagnetic final state interaction in the SM and
contribution due to charged-Higgs exchange diagrams in the framework of the
Weinberg three doublets model. It is shown that at the one-loop level of the SM
the muon transverse polarization, $P_T^{SM}$, varies within $(0.0 \div 1.1
\cdot 10^{-3})$ in the Dalitz plot region. Averaged value of the muon
polarization, $<P^{SM}_T>$, in the kinematic region of $E_\gamma \geq 20$ MeV
is equal to $4.76\cdot 10^{-4}$. In the case of the model with three Higgs
doublets the muon transverse polarization is calculated as a function of
charged Higgs masses, imaginary part of the Yukawa coupling constants product
and vacuum expectation values of the Higgs doublets. The averaged value of the
transverse polarization in the Weinberg model is $<P^{Higgs}_T > = - 6.62 \cdot
10^{-5}$. Perspectives to probe the effect caused by the charged Higgs exchange
diagrams in planned kaon experiments are discussed.
