The invention relates to the soybean variety designated A1037403. Provided by the invention are the seeds, plants and derivatives of the soybean variety A1037403. Also provided by the invention are tissue cultures of the soybean variety A1037403 and the plants regenerated therefrom. Still further provided by the invention are methods for producing soybean plants by crossing the soybean variety A1037403 with itself or another soybean variety and plants produced by such methods.