$\def\ov{\overline}$As Wlodek Kuperberg mentioned in the comments, it is reasonable to assume that no five points lie on a sphere. Under this condition, I claim that 6 points suffice.

**1.** *We may choose a sphere passing through four given points and separatinf the other two.*

To prove this, choose a face $ABC$ of the convex hull of given points, start with a very large sphere passing through $A$, $B$, $C$ and containing no other points inside it. Blow it down and then up again till it becomes a large sphere on the other side containing the other three points. It will take these points one by one, so when it passes through the second point we catch the required moment.

**2.** Now let $A$, $B$, $C$, $D$ be our points on the sphere $\Omega$, $E$ be the point outside $\Omega$, and $F$ be the point inside $\Omega$. Let the line $EF$ meet the sphere at $X$ and $Y$ (with the order $E$, $X$, $F$, $Y$ on the line). Then there exists a point $R$ on the segment $XF$ such that $\ov{RE}\cdot \ov{RF}=\ov{RX}\cdot \ov{RY}$; notice that $R$ lies inside $\Omega$.

This point is magical in the following sense: *If $AR$ meets $\Omega$ at $A'$, then the points $A$, $E$, $F$, and $A'$ are concyclic* (since $\ov{RA}\cdot \ov{RA'}=\ov{RX}\cdot \ov{RY}=\ov{RE}\cdot \ov{RF}$). Thus, *If the plane $BCD$ separates $A$ and $A'$, then the circle $\omega=(AEF)$ is linked with $(BCD)$* (indeed, the arc $AFA'$ of $\omega$ crosses the disk bounded by $(BCD)$ exactly once, while the other arc $AEA'$ does not). In this case we say that the point $A$ *works*.

**3.** So it remains to show that one of the four points $A$, $B$, $C$, $D$ works. If $R$ lies inside the tetrahedron $ABCD$, then every its vertex works. Otherwise, $R$ is (non-strictly) separated from this tetrahedron by a plane containing one of its faces (say, $BCD$). In this case, the (infinite) cone with apex $A$ and base $(BCD)$ contains $R$ (since it contains the intersection of the interior of $\Omega$ with the corresponding half-space of $BCD$), so $A$ works.

Yes. – Joseph O'Rourke Aug 26 at 23:09