Aims: We aim to monitor the optical linear polarimetric signal of the magnetized, rapidly rotating M8.5 dwarf TVLM 513-46546. Methods: R- and I-band linear polarimetry images were collected with the Andalucía Faint Object Spectrograph and Camera (ALFOSC) instrument of the 2.56-m Nordic Optical Telescope (NOT) on two consecutive nights, covering about 0.5 and four rotation cycles in the R and I filters, respectively. We also obtained simultaneous intensity curves by means of differential photometry. The typical precision of the data is ±0.46% (R), ±0.35% (I) in the linear polarization degree and ±9 mmag (R), ±1.6 mmag (I) in the differential intensity curves. Results: Strong and variable linear polarization is detected in the R and I filters, with values of maximum polarization (p∗ = 1.30 ± 0.35 %), that are similar for both bands. The intensity and the polarimetric curves present a sinusoid-like pattern with a periodicity of ~1.98 h, which we ascribe to structures in the surface of TVLM 513-46546 synchronized with its rotation. We found that the peaks of intensity and polarimetric curves occur with a phase difference of 0.18 ± 0.01 and that the maximum of the linear polarization occurs nearly half a period (0.59 ± 0.03) after the radio pulse. We discuss different scenarios to account for the observed properties of the light curves. Appendix A is available in electronic form at http://www.aanda.org