Linear polarization can be used as a probe of the existence of atmospheric condensates in ultracool dwarfs. Models predict that the observed linear polarization increases with the degree of oblateness, which is inversely proportional to the surface gravity. We aimed to test the existence of optical linear polarization in a sample of bright young brown dwarfs, with spectral types between M6 and L2, observable from the Calar Alto Observatory, and cataloged previously as low gravity objects using spectroscopy. Linear polarimetric images were collected in I and R band using CAFOS at the 2.2-m telescope in Calar Alto Observatory (Spain). The flux ratio method was employed to determine the linear polarization degrees. With a confidence of 3σ, our data indicate that all targets have a linear polarimetry degree in average below 0.69�per�cent in the I band, and below 1.0�per�cent in the R band, at the time they were observed. We detected significant (i.e. P/σ ≥ 3) linear polarization for the young M6 dwarf 2MASS J04221413+1530525 in the R band, with a degree of p^*�=�0.81���0.17�per�cent.