We present high-precision linear polarization observations of four bright hot Jupiter systems ($\tau$ Boo, HD 179949, HD 189733 and 51 Peg) and use the data to search for polarized reflected light from the planets. The data for 51 Peg are consistent with a reflected light polarization signal at about the level expected with 2.8$\sigma$ significance and a false alarm probability of 1.9 per cent. More data will be needed to confirm a detection of reflected light in this system. HD 189733 shows highly variable polarization that appears to be most likely the result of magnetic activity of the host star. This masks any polarization due to reflected light, but a polarization signal at the expected level of $\sim$20 ppm cannot be ruled out. $\tau$ Boo and HD 179949 show no evidence for polarization due to reflected light. The results are consistent with the idea that many hot Jupiters have low geometric albedos. Conclusive detection of polarized reflected light from hot Jupiters is likely to require further improvements in instrument sensitivity.