We have used the seven-year Wilkinson Microwave Anisotropy Probe (WMAP) data in order to update the measurements of the intensity signal in the G159.6-18.5 region within the Perseus molecular complex and to set constraints on the polarization level of the anomalous microwave emission in the frequency range where this emission is dominant. At 23, 33, and 41 GHz, we obtain upper limits on the fractional linear polarization of 1.0%, 1.8%, and 2.7%, respectively (with a 95% confidence level). These measurements rule out a significant number of models based on magnetic dipole emission of grains that consist of a simple domain as responsible for the anomalous emission. When combining our results with the measurement obtained with the COSMOSOMAS experiment at 11 GHz, we find consistency with the predictions of the electric dipole and resonance relaxation theory at this frequency range.