Understanding the droplet coalescence/merging is vital for many areas of microfluidics such as biochemical reactors, drug delivery, inkjet printing, oil recovery, etc. In the present study, we carried out numerical simulations of two magnetic droplets suspended in a nonmagnetic fluid matrix and coalescing under the influence of an external magnetic field. We observed that the applied magnetic field played a key role in the merging dynamics of the magnetic droplets. When the two droplets make the first contact with each other, a microscopic liquid bridge forms between the two and grows rapidly in the lateral direction until it coalesces into one. The temporal evolution of the neck radius with the onset of coalescence gives the growth rate of the liquid bridge. In the present study, parameters such as magnetic Bond number, magnetic susceptibility, and the viscosity ratio of the outer ambient fluid to droplet fluid were varied, and the bridge radius growth rate was assessed. The current study aims to discern how parameters such as magnetic Bond number, magnetic susceptibility, and the viscosity ratio influence the growth rate of the liquid bridge that forms between the droplets during coalescence. It is observed that the growth rate of the bridge radius is significantly affected by the change in magnetic Bond number and magnetic susceptibility for a high viscosity ratio. In contrast, for low viscosity ratio cases, the influence of magnetic Bond number and magnetic susceptibility on the rate of bridge growth is negligible. This unveils the implicit relationship among the three aforementioned parameters. Furthermore, we observe that the spatial structure of the neck region varied with the viscosity ratio and affected the rate of expansion of the neck radius. This study reveals how a magnetic influence can manipulate the structure of the neck region of two merging droplets and in turn affect the growth rate.