I am not a mathematician, but I know that a sphere projected on the image plane becomes an ellipse under perspective transformation.
I don't know whether the ray starting from the center of projection passing through the center of the ellipse also passes through the center of the sphere.
According to this image, I would say no. But I am not sure. If not, how can I find the ray which passes through the center of a sphere given the 2D image coordinates of the ellipse? The camera is calibrated and I know the size of the sphere.
UPDATE
Here is a visual representation of David K's excellent answer:
You can construct a line through the center of the sphere if you know the center of the projection and if you can identify the axes of the ellipse.
Let $O$ be the center of the projection and let $P$ and $Q$ be the endpoints of the major axis of the ellipse. Construct the lines $OP$ and $OQ$. These lines lie along the surface of the cone. The plane containing these lines also contains the axis of the cone. Bisect the angle $\angle POQ$. Then the angle bisector is the axis of the cone and passes through both the center of the projection and the center of the sphere.
The angle bisector of $\angle POQ$ intersects the major axis of the ellipse at a point $M$ that divides the major axis into two segments whose lengths have the ratio $\lvert MP\rvert:\lvert MQ\rvert = \lvert OP\rvert:\lvert OQ\rvert$. This implies that $M$ is the center of the ellipse only if $\lvert OP\rvert = \lvert OQ\rvert$, in which case the ellipse is a circle.
If you know the radius and location of the sphere then it is possible to find the point $M$ using that information.
If you know only the ellipse then it is not possible to identify the point $M$ where the axis of the cone intersects the plane of the ellipse. This is because there are infinitely many points that could be the center of a projection mapping a sphere to the given ellipse. See the answers to From ellipse equation to circular cone axis.