We develop here a new approach for the relativistic modeling of the photons moving into a quasi-Minkowskian spacetime, where the metric is generated by an arbitrary N-body distribution within an isolated solar system. Our model is built on the prescriptions of the theory of general relativity and leaves the choices of the metric, as well as that of the motion of the observer, arbitrary. Adopting a quasi-Minkowskian expression of the metric accurate to order (v/c)², we conduct a thorough numerical test campaign to verify correctness and reliability of the model equations. The test results show that the model behaves according to predictions. Specifically, comparisons to true (simulated) data demonstrate that stellar distances are reconstructed up to the specified level of accuracy. Although the (v/c)² approximation is not always sufficient for its application to future astrometric experiments, which require modeling to (v/c)³, this work serves also as a natural test ground for the higher order model, whose formulation is now close to completion and will be presented in a forthcoming paper.