In a locally contractible topological space $X$ is it possible at each point $x$ to find a local basis of contractible sets $U_i\ni x$ such that the closure of each set $\overline{U_i} \subset X$ is also locally contractible?

More precisely, for this question we may assume $X$ is compact and is an ANR (for the class of separable metric spaces), we can even assume that $X$ is embedded as a subspace of $\mathbb{R}^n$ if that makes the question easier.