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<jats:title>Abstract</jats:title> <jats:p>Polarized foreground emission is a potential contaminant of attempts to measure the fluctuation power spectrum of highly redshifted 21 cm H <jats:sc>i</jats:sc> emission from the epoch of reionization. Using the Donald C. Backer Precision Array for Probing the Epoch of Reionization, we present limits on the observed power spectra of all four Stokes parameters in two frequency bands, centered at 126 MHz (<jats:italic>z</jats:italic> = 10.3) and 164 MHz (<jats:italic>z</jats:italic> = 7.66), for a three-month observing campaign of a deployment involving 32 antennas, for which results on the unpolarized power spectrum have been reported at <jats:italic>z</jats:italic> = 7.7 (by Parsons et al.) and at <jats:inline-formula> <jats:tex-math> <?CDATA $7.5\lt z\lt 10.5$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjaa5970ieqn1.gif" xlink:type="simple" /> </jats:inline-formula> (by Jacobs et al.). The power spectra in this paper are processed in the same way as by those authors, and show no definitive detection of polarized power. This nondetection is consistent with what is known about polarized sources, combined with the suppression of polarized power by fluctuations in the ionospheric rotation measure, which can strongly affect Stokes <jats:italic>Q</jats:italic> and <jats:italic>U</jats:italic>. We are able to show that the net effect of polarized leakage is a negligible contribution at the levels of the limits reported by Parsons et al. and Jacobs et al.</jats:p>