More specifically, letting $\mathbb S$ be the set $\{\kern.5mm\exp({\rm i}\,t):0\le t<2\pi\kern.6mm\}$ with the induced topology from the complex plane, let $\mathbb T^{\kern.6mm\infty}$ be the product topological space of denumerably many of copies of $\mathbb S$. E.g. from page 15 in this paper we know that all separable infinite-dimensional Fréchet spaces are homeomorphic. So the question is whether $\mathbb T^{\kern.6mm\infty}$ is homeomorphic to an open subset in some separable infinite-dimensional Fréchet space.
Edit. After reading Igor Belegradek's comment, I now realize what I should have realized already before posting the question. Namely, the motivation behind the question was whether $\mathbb T^{\kern.6mm\infty}$ could be made a manifold modelled on Fréchet spaces with charts defined on open subsets. These sets are $\sigma$−compact, and hence cannot be homeomorphic to open subsets in any infinite-dimensional Baire topological vector space. So there is no hope to turn $\mathbb T^{\kern.6mm\infty}$ into a manifold modelled on Fréchet spaces. The question can be closed.