This paper introduces the Visual Inverse Kinematics problem (VIK) to fill the gap between robot Inverse Kinematics (IK) and visual servo control. Different from the IK problem, the VIK problem seeks to find robot configurations subject to vision-based constraints, in addition to kinematic constraints. In this work, we develop a formulation of the VIK problem with a Field of View (FoV) constraint, enforcing the visibility of an object from a camera on the robot. Our proposed solution is based on the idea of adding a virtual kinematic chain connecting the physical robot and the object; the FoV constraint is then equivalent to a joint angle kinematic constraint. Along the way, we introduce multiple vision-based cost functions to fulfill different objectives. We solve this formulation of the VIK problem using a method that involves a semidefinite program (SDP) constraint followed by a rank minimization algorithm. The performance of this method for solving the VIK problem is validated through simulations.