It is shown that the above-threshold electron de Broglie waves, generated by an intense laser pulse at a metal surface are interfering to yield attosecond electron pulses. This interference of the de Broglie waves is an analog on of the superposition of high harmonics generated from rare gas atoms, resulting in trains of attosecond light pulses. Our model is based on the Floquet analysis of the inelastic electron scattering on the oscillating double-layer potential, generated by the incoming laser field of long duration at the metal surface. Owing to the inherent kinematic dispersion, the propagation of attosecond de Broglie waves in vacuum is very different from that of attosecond light pulses, which propagate without changing shape. The clean attosecond structure of the current at the immediate vicinity of the metal surface is largely degraded due to the propagation, but it partially recovers at certain distances from the surface. Accordingly, above the metal surface, there exist “collapse bands,” where the electron current is erratic or noise-like, and there exist “revival layers,” where the electron current consist of ultrashort pulses of about 250 attosecond durations in the parameter range we considered. The maximum value of the current densities of such ultrashort electron pulses has been estimated to be on order of couple of tenth of mA/cm2. The attosecond structure of the electron photocurrent can perhaps be used for monitoring ultrafast relaxation processes in single atoms or in condensed matter.