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An analytical theory is developed that describes acoustic microstreaming produced by the interaction of an oscillating gas bubble with a viscoelastic particle. The bubble is assumed to undergo axisymmetric oscillation modes, which can include radial oscillation, translation and shape modes. The oscillations of the particle are excited by the oscillations of the bubble. No restrictions are imposed on the ratio of the bubble and the particle radii to the viscous penetration depth and the separation distance, as well as on the ratio of the viscous penetration depth to the separation distance. Capabilities of the developed theory are illustrated by computational examples. The shear stress produced by the acoustic microstreaming on the particle surface is calculated. It is shown that this stress is much higher than the stress predicted by the formula of Nyborg, which is commonly used to evaluate the time-averaged shear stress produced by a bubble on a rigid wall.