In the power scaling of monolithic fiber lasers, the fiber nonlinear effects and transverse mode instability are main limitations. The tapered gain fiber has a longitudinally varying mode area, which has the advantage of mitigating fiber nonlinear effects. However, the transverse mode instability (TMI) was seldom reported in the tapered fiber lasers at high average power levels. In this work, we have constructed a monolithic tapered ytterbium-doped fiber laser oscillator and investigated the laser oscillator performance with respective 976 nm and 915 nm pump, especially on the aspects of the TMI. The double cladding tapered ytterbium-doped fiber has a narrow end of ~20/400 μm and a wide end of ~30/600 μm. Fiber Bragg gratings (FBG) are respectively inscribed on double cladding fibers with core/inner cladding diameter of 20/400 μm and 30/400 μm to match with the narrow and wide end of the tapered ytterbium-doped fiber. When 915 nm pump is employed, the TMI occurs at the output power of ~1350 W. The output power is further scaled to a maximum of 1720 W. The M2 factor of the output laser is ~2.1 and the full width at half maximum (FWHM) of the signal laser is ~3.6 nm. To the best of our knowledge, this is the highest average power for the tapered ytterbium-doped fiber lasers.