High power fiber lasers with high beam quality have been widely employed in applications of industrial manufacture. However, the power scaling of the fiber lasers with high beam quality are limited by the fiber nonlinear effects and the transverse mode instability. It is promising to simultaneously mitigate the fiber nonlinear effects and transverse mode instability by employing large mode area gain fiber with novel structure. In this letter, we report a 6 kW fiber laser with high beam quality, which is realized based on home-made ytterbium-doped fiber with gradually varying spindle-shape structure. The fiber laser employs the master oscillation power amplification structure. In the stage of laser amplifier, spindle-shape ytterbium-doped fiber is bidirectionally pumped by laser diodes with 981 nm wavelength. At the total pump power of 7.68 kW, the maximum output power reaches 6.02 kW with a beam quality M² factor of 1.9. By optimizing the manufacture techniques and structure parameters of the spindle-shape ytterbium-doped fiber, it is promising to achieve fiber lasers with higher power and nearly single mode beam quality.