Anomalous Microwave Emission (AME) has been previously studied in two well-known molecular clouds and is thought to be due to electric dipole radiation from small spinning dust grains. It is important to measure the polarization properties of this radiation both for component separation in future cosmic microwave background experiments and also to constrain dust models. We have searched for linearly polarized radio emission associated with the $ρ$ Ophiuchi and Perseus molecular clouds using {\it WMAP} 7-year data. We found no significant polarization within an aperture of $2^{∘}$ diameter. The upper limits on the fractional polarization of spinning dust in the $ρ$ Ophiuchi cloud are 1.7%, 1.6% and 2.6% (at 95% confidence level) at K-, Ka- and Q-bands, respectively. In the Perseus cloud we derived upper limits of 1.4%, 1.9% and 4.7%, at K-, Ka- and Q-bands, respectively; these are similar to those found by López-Caraballo et al. If AME at high Galactic latitudes has a similarly low level of polarization, this will simplify component separation for CMB polarization measurements. We can also rule out single domain magnetic dipole radiation as the dominant emission mechanism for the 20-40 GHz. The polarization levels are consistent with spinning dust models. ; Comment: Accepted in MNRAS as a letter - added extra sentence. 5 pages, 2 figures, 1 table