  We study a connection between a multivariable version of the
Goodwillie-Weiss' calculus of functors and derived mapping spaces of k-fold
bimodules over a family of operads. As our main application, under the
assumption $d_{i}+3\leq n$ for all $i\in \{1,\ldots,k\}$, we produce explicit
deloopings of high-dimensional spaces of string links modulo immersions from
$\sqcup_{i}\mathbb{R}^{d_{i}}$ to $\mathbb{R}^{n}$ and their polynomial
approximations.
