A high-brightness electron beam source for a microfocus X-ray tube has been fabricated using a carbon-nanotube (CNT) field emitter. The electron source consists of cathode that includes a CNT field emitter, a beam-extracting grid, and an anode that accelerates that electron beam. The microfocus X-ray tube requires an electron beam with the diameter of less than 5 $mu$m and beam current of higher than 30 $mu$A at the position of the X-ray target. To satisfy the requirements, the geometries of the field emitter tips and the electrodes of the gun was optimized by calculating the electron trajectories and beam spatial profile with EGUN code. The CNT tips were fabricated with successive steps: a tungsten wire with the diameter of 200 $mu$m was chemically etched and was subsequently coated with CNTs by chemical vapor deposition. The experiments of electron emission at the fabricated CNT tips were performed. The design characteristics and basic experimental results of the electron source are reported.