No. The space of homeomorphisms of a compact manifold is locally contractible:

A. V. Černavskiı̆. Local contractibility of the group of homeomorphisms of a manifold. Mat. Sb. (N.S.), 79 (121):307–356, 1969.

So if there were such a sequence then for large enough $i$ the diffeomorphism $f_i$ would be topologically isotopic to $f$. But there are homomorphisms which are not isotopic to diffeomorphisms.

For example, let $\Sigma$ be a smooth homotopy $d$-sphere which does not have order 2 in the group $\Theta_d$ of such (e.g. Milnor's exotic 7-sphere). As $\Sigma$ is homeomorphic to $S^d$ (by the topological Poincare conjecture) it admits an orientation-reversing homeomorphism $f : \Sigma \to \Sigma$. But this $f$ cannot even be homotopic to a diffeomorphism, for if it was it would mean that $[\Sigma] = - [ \Sigma] \in \Theta_d$.

No. The space of homeomorphisms of a compact manifold is locally contractible:

A. V. Černavskiı̆. Local contractibility of the group of homeomorphisms of a manifold. Mat. Sb. (N.S.), 79 (121):307–356, 1969.

So if there were such a sequence then for large enough $i$ the diffeomorphism $f_i$ would be topologically isotopic to $f$. But there are homomorphisms which are not isotopic to diffeomorphisms.

For example, let $\Sigma$ be a smooth homotopy $d$-sphere which does not have order 2 in the group $\Theta_d$ of such (e.g. Milnor's exotic 7-sphere). As $\Sigma$ is homeomorphic to $S^d$ (by the topological Poincare conjecture) it admits an orientation-reversing homeomorphism $f : \Sigma \to \Sigma$. But this $f$ cannot even be homotopic to a diffeomorphism, for if it were it would mean that $[\Sigma] = - [ \Sigma] \in \Theta_d$.

No. The space of homeomorphisms of a compact manifold is locally contractible, so if there were such a sequence then for large enough $i$ the diffeomorphism $f_i$ would be topologically isotopic to $f$. But there are homomorphisms which are not isotopic to diffeomorphisms.

No. The space of homeomorphisms of a compact manifold is locally contractible:

A. V. Černavskiı̆. Local contractibility of the group of homeomorphisms of a manifold. Mat. Sb. (N.S.), 79 (121):307–356, 1969.

So if there were such a sequence then for large enough $i$ the diffeomorphism $f_i$ would be topologically isotopic to $f$. But there are homomorphisms which are not isotopic to diffeomorphisms.

For example, let $\Sigma$ be a smooth homotopy $d$-sphere which does not have order 2 in the group $\Theta_d$ of such (e.g. Milnor's exotic 7-sphere). As $\Sigma$ is homeomorphic to $S^d$ (by the topological Poincare conjecture) it admits an orientation-reversing homeomorphism $f : \Sigma \to \Sigma$. But this $f$ cannot even be homotopic to a diffeomorphism, for if it was it would mean that $[\Sigma] = - [ \Sigma] \in \Theta_d$.