  We investigate the possibility of oscillon formation during the preheating
phase of asymmetric inflationary potentials. We analytically establish the
existence of oscillon-like solutions for the Klein-Gordon equation for a
polynomial potential of the form $V(\phi)=\frac{1}{2}\phi^2+A\phi^3+B\phi^4$
using the small amplitude analysis, which naturally arises as a Taylor
expansion of the $\alpha$-attractor E-model for $\phi\ll M_\text{pl}$ and
$\alpha\sim\mathcal{O}(1)$. We perform a detailed numerical analysis to study
the formation of nonlinear structures in the $\alpha$-attractor E-model using
the publicly available lattice simulation code
$\mathcal{C}\text{osmo}\mathcal{L}\text{attice}$ for parameters in the range
$10^{-5}\lesssim\alpha\lesssim 5\times 10^{-4}$. We find the backreaction of
the field fluctuations onto the evolution of the homogeneous inflaton
condensate to be significant for $\alpha\lesssim 2\times 10^{-4}$ for which we
observe the formation of highly nonlinear structures with average equation of
state $w\simeq 0$. These nonlinear structures maybe interpreted as oscillons,
providing evidence that they can form during the inflaton oscillations around
an asymmetric potential and are found to be present for the entirety of the
runtime of our simulations, comprising $\gtrsim 40\%$ of the total energy
density.
