Let me add another answer. This answer stems from the fruitful discussion on MathOverflow.

**Theorem.** There is a topological embedding $\iota:\mathbb{S}^1\to\mathbb{S}^5$ such that $\pi_3(\mathbb{S}^5\setminus\iota(\mathbb{S}^1))=0$.
Therefore, no
$3$-sphere can be linked with $\iota(\mathbb{S}^1)$.

**Proof.**
It is well known that there are $3$-dimensional integer homology spheres whose universal cover is $\mathbb{R}^3$. For example, there are particular constructions in [1,2,5] of hyperbolic integer homology spheres. Note that the universal cover of a hyperbolic $3$-manifold is the hyperbolic space that is homeomorphic to $\mathbb{R}^3$.
Other examples are listed in Homology sphere with $\mathbb{R}^3$ as the universal cover.
Let $\mathcal M$ be such an integer homology sphere. Since the universal cover of $\mathcal M$ is contractible, $\pi_3(\mathcal M)=0$. According to the celebrated theorem of Cannon and Edwards [3,4], the double suspension of an integer homology sphere is homeomorphic to a topological sphere. Let $h:S^2{\mathcal M}\to\mathbb{S}^5$ be such a homeomorphism.
$\mathcal M$ is a deformation retract of the complement of the vertices of the suspension $S\mathcal M$.
Therefore, $\mathcal M$ is also a deformation retract of the complement of the suspension of the vertices in $S^2\mathcal M$. Denote the suspension of the vertices by $X$, so $\mathcal M$ is a deformation retract of $S^2{\mathcal M}\setminus X$ and hence $\pi_3(S^2\mathcal{M}\setminus X)=0$.
$X$ is homeomorphic to $\mathbb{S}^1$. If $g:\mathbb{S}^1\to X$ is a homeomorphism, then $\iota=h\circ g:\mathbb{S}^1\to\mathbb{S}^5$ is a topological embedding and clearly $\pi_3(\mathbb{S}^5\setminus\iota(\mathbb{S}^1))=\pi_3(\mathbb{S}^5\setminus h(X))=\pi_3(S^2{\mathcal M}\setminus X)=0$.
The proof is complete. $\Box$

**[1] J. Baldwin, J., S. Sivek,**
Stein fillings and $SU(2)$ representations. *Geom. Topol.* 22 (2018), 4307-4380.

**[2] J. Brock, N. M. Dunfield,**
Injectivity radii of hyperbolic integer homology $3$-spheres.
*Geom. Topol.* 19 (2015), 497-523.

**[3] J. W. Cannon,** Shrinking cell-like decompositions of manifolds. Codimension three. *Ann. of Math.* 110 (1979), 83-112.

**[4] R. D. Edwards,** The topology of manifolds and cell-like maps. Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 111–127, Acad. Sci. Fennica, Helsinki, 1980.

**[5] J. Hom, T. Lidman,**
A note on surgery obstructions and hyperbolic integer homology spheres.
*Proc. Amer. Math. Soc.* 146 (2018), 1363-1365.