We present linear polarization measurements of nearby FGK dwarfs to parts-per-million ( ppm ) precision . Before making any allowance for interstellar polarization , we found that the active stars within the sample have a mean polarization of 28.5 \pm 2.2 ppm while the inactive stars have a mean of 9.6 \pm 1.5 ppm . Amongst inactive stars we initially found no difference between debris disk host stars ( 9.1 \pm 2.5 ppm ) and the other FGK dwarfs ( 9.9 \pm 1.9 ppm ) . We develop a model for the magnitude and direction of interstellar polarization for nearby stars . When we correct the observations for the estimated interstellar polarization we obtain 23.0 \pm 2.2 ppm for the active stars , 7.8 \pm 2.9 ppm for the inactive debris disk host stars and 2.9 \pm 1.9 ppm for the other inactive stars . The data indicates that whilst some debris disk host stars are intrinsically polarized most inactive FGK dwarfs have negligible intrinsic polarization , but that active dwarfs have intrinsic polarization at levels ranging up to \sim 45 ppm . We briefly consider a number of mechanisms , and suggest differential saturation of spectral lines in the presence of magnetic fields is the best able to explain the polarization seen in active dwarfs . The results have implications for current attempts to detect polarized reflected light from hot Jupiters by looking at the combined light of the star and planet .