The origin of the Fermi bubble, which constitute two gamma-ray emitting lobes above and below the Galactic plane, remains unclear. The possibility that this Fermi bubble gamma-ray emission originates from hadronic cosmic rays advected by a subsonic Galactic outflow, or breeze, is explored. The simulation of a breeze solution and subsequent cosmic ray transport is carried out using the hydrodynamical code, PLUTO, in combination with a cosmic ray transport code. The Galactic outflow model obtained is found to be compatible with both inferences of the decelerating outflow velocity profile of the gas in the Fermi bubble region, and evidence for the presence of a large amount of hot ionised gas out in the Galactic halo region. Although simple, this model is found to be able to reproduce the observed Fermi-LAT energy flux at high Galactic latitudes.