In this work, a tunable single-frequency fiber laser with asymmetric compound ring cavity is designed. Fiber ring resonator, fiber optic tunable filter, and fiber ring filter are respectively embedded into the main fiber ring cavity. An efficient and stable tunable single-frequency fiber laser from 1020 nm to 1090 nm is achieved based on the principle of the vernier effect. A 980 nm semiconductor laser is used as the pump source, when the pump power is 200 mW, the output linewidth of the single-frequency laser measured by the delayed self-heterodyne method is 3.325 kHz, and the optical signal-to-noise ratio is greater than 50 dB, with 5 min as the interval, the measured output laser wavelength stability is 0.02 nm within 40 min.