Using high-resolution spectropolarimetric data recorded with the Swedish 1 m Solar Telescope, we have identified several instances of granular lanes traveling into granules. These are believed to be the observational signature of underlying tubes of vortical flow with their axis oriented parallel to the solar surface. Associated with these horizontal vortex tubes, we detect in some cases a significant signal in linear polarization, located at the trailing dark edge of the granular lane. The linear polarization appears at a later stage of the granular lane development, and is flanked by patches of circular polarization. Stokes inversions show that the elongated patch of linear polarization signal arises from the horizontal magnetic field aligned with the granular lane. We analyze snapshots of a magnetohydrodynamic numerical simulation and find cases in which the horizontal vortex tube of the granular lane redistributes and transports the magnetic field to the solar surface causing a polarimetric signature similar to what is observed. We thus witness a mechanism capable of transporting magnetic flux to the solar surface within granules. This mechanism is probably an important component of the small-scale dynamo supposedly acting at the solar surface and generating the quiet-Sun magnetic field.