Attosecond electron bunches have wide application prospects in free-electron laser injection, attosecond X/γ-ray generation, ultrafast physics, etc. Nowadays, there is one notable challenge in the generation of high-quality attosecond electron bunch, i.e., how to enhance the electron bunch density. Using theoretical analysis and three-dimensional particle-in-cell simulations, we discovered that a relativistic vortex laser pulse interacting with near-critical density plasma can not only effectively concentrate the attosecond electron bunches to over critical density, but also control the duration and density of the electron bunches by tuning the intensity and carrier-envelope phase of the drive laser. It is demonstrated that this method can efficiently produce attosecond electron bunches with a density up to 300 times of the original plasma density, peak divergence angle of less than 0.5∘, and duration of less than 67 attoseconds. Furthermore, by using near-critical density plasma instead of solid targets, our scheme is potential for the generation of high-repetition-frequency attosecond electron bunches, thus reducing the requirements for experiments, such as the beam alignment or target supporter.