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The effect of nonuniform viscosity on the swimming velocity of a free swimmer at zero Reynolds number is examined. Using the generalized reciprocal relation for Stokes flow with nonuniform viscosity, we formulate the locomotion problem in a fluid medium with spatially varying viscosity. Assuming the limit of small variation in the viscosity of the fluid as a result of nonuniform distribution of nutrients around a swimmer, we derive a perturbation model to calculate the changes in the swimming performance of a spherical swimmer as a result of position-dependent viscosity. The swimmer is chosen to be a spherical squirmer with a steady tangential motion on its surface modeling ciliary motion. The nutrient concentration around the body is described by an advection-diffusion equation. The roles of the surface stroke pattern, the specific relationship between the nutrient and viscosity, and the Peclet number of the nutrient in the locomotion velocity of the squirmer are investigated. Our results show that for a pure treadmill stroke, the velocity change is maximum at the limit of zero Peclet number and monotonically decreases toward zero at very high Peclet number. When higher surface stroke modes are present, larger modification in swimming velocity is captured at high Peclet number where two mechanisms of thinning the nutrient boundary layer and appearance of new stagnation points along the surface of squirmer are found to be the primary reasons behind the swimming velocity modifications. It is observed that the presence of nonuniform viscosity allows for optimal swimming speed to be achieved with stroke combinations other than pure treadmill.