  We search for the $Z_1(4050)^+$ and $Z_2(4250)^+$ states, reported by the
Belle Collaboration, decaying to $\chi_{c1} \pi^+$ in the decays $\bar B^0 \to
\chi_{c1} K^- \pi^+$ and $B^+ \to \chi_{c1} K^0_S \pi^+$ where $\chi_{c1} \to
\jpsi \gamma$. The data were collected with the BaBar detector at the SLAC
PEP-II asymmetric-energy $e^+e^-$ collider operating at center-of-mass energy
10.58 GeV, and correspond to an integrated luminosity of 429 fb$^{-1}$. In this
analysis, we model the background-subtracted, efficiency-corrected
$\chi_{c1}\pi$ mass distribution using the $K \pi$ mass distribution and the
corresponding normalized $K \pi$ Legendre polynomial moments, and then test the
need for the inclusion of resonant structures in the description of the
$\chi_{c1}\pi$ mass distribution. No evidence is found for the $Z_1(4050)^+$
and $Z_2(4250)^+$ resonances, and 90% confidence level upper limits on the
branching fractions are reported for the corresponding $B$-meson decay modes.
