We present the results of polarization measurements of stars projected on four dense cores that are found to harbor Very Low Luminosity Objects detected by the Spitzer Space telescope. The observed dense cores are IRAM 04191+1522, L1521F, L673-7 and L1014. In IRAM 04191+1522, the separation between the magnetic field direction inferred from the optical polarization and the outflow direction is 84 degree while the separation between the magnetic field inferred from the submillimeter polarization and the outflow direction is ~ 3 degree. IRAM 04191+1522 exhibits a wind-blown morphology conspicuous in the WISE 12μm image. The direction of the maximum velocity gradient in IRAM 04191+1522 is found to be in a direction roughly perpendicular to the direction of the magnetic field inferred from the submillimeter observations. In L1521F the offset between the outflow direction and the magnetic field from our optical polarization is found to be ~ 48 degree. In L673-7 dense core, the magnetic field lines are found to be more chaotic with a slight hint of the outflow being parallel to the field towards the northern parts of the cloud. The offset between the magnetic field orientation and the outflow is found to be ~ 8 degree. In L1014, the separation between the magnetic field orientation and the outflow is found to be ~ 15 degree. The minor axis of the cloud is almost parallel to the magnetic field orientation at the periphery of the cloud.