The definition of an element of the fundamental group of a space $X$ based at point $p$ is $$f:[0,1]\rightarrow X,\quad f(0)=p=f(1),$$ defined up to homotopy. This homotopy allows self-intersection, so that there is no sense in thinking about images of loops as knots. Suppose we were to constrain the homotopy class by disallowing self-intersection. Then for each element of the ordinary fundamental group, we would have a copy of every possible knot.
My feeling is that this would get us nowhere, but I can't help but ask if anyone has thought about this and if there might be an interesting application. For instance, how about the complement of a sphere that has some infinite set of knots drilled out of it?