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Recently proposed cosmological $alpha$-attractors are observationally favored due to the asymptotic flatness of the potential. It is known that since the flatness induces the negative pressure, coherent oscillation of the inflaton field could fragment into the quasi-stable localized objects called oscillons. We investigated the possibility of oscillon formation in E-models of $alpha$-attractors. Using the linear analysis and the lattice simulations, we find that the instability sufficiently grows against the cosmic expansion and inflaton fragments into the oscillons for $alphalesssim10^{-3}$, where $alpha$ is the model parameter of the $alpha$-attractors.