The construction of a two-dimensional magneto–optical trap with hollow cooling and pushing (2D-HP MOT) is reported in detail, and a velocity-tunable cold atomic beam produced by this 2D-HP MOT is demonstrated. The magneto–optical trap system, which is constructed by a transparent quartz tube, is low in price, easy to fabricate and assemble, and convenient for atomic trapping and detection. The mean axial velocity of the cold atomic beam can be tuned from 4.5 to 8 m/s, while the atomic flux remains at a level of ${10}^{10}\mathrm{atoms}/\mathrm{s}$. This cold atomic beam source can be applied in the areas of high-precision measurements based on cold atoms.