The radio quiet quasar SDSS J1240+1455 lies at a redshift of z = 3.11, is surrounded by a Ly alpha blob (LAB), and is absorbed by a proximate damped Ly alpha system. In order to better define the morphology of the blob and determine its emission mechanism, we gathered deep narrow-band images isolating the Ly alpha line of this object in linearly polarized light. We provide a deep intensity image of the blob, showing a filamentary structure extending up to 16 '' (or 122 physical kpc) in diameter. No significant polarization signal could be extracted from the data, but 95% probability upper limits were defined through simulations. They vary between similar to 3% in the central 0.75 '' disk (after subtraction of the unpolarized quasar continuum) and similar to 10% in the 3.8 - 5.5 '' annulus. The low polarization suggests that the Ly alpha photons are emitted mostly in situ, by recombination and de-excitation in a gas largely ionized by the quasar ultraviolet light, rather than by a central source and scattered subsequently by neutral hydrogen gas. This blob shows no detectable polarization signal, contrary to LAB1, a brighter and more extended blob that is not related to the nearby active galactic nucleus (AGN) in any obvious way, and where a significant polarization signal of about 18% was detected.