There are several synonyms for the phrase "border upon," which refers to something being near or close to something else. Some alternatives include: adjoin, abut, touch, neighbor, approach, and verge on. These words can all be used to describe the proximity of one thing to another, whether it's physical or metaphorical. For example, "The park abuts the river," or "His behavior verges on offensive." Each of these words carries a slightly different connotation, so it's important to choose the one that best fits the context of the sentence. Nonetheless, all of these synonyms can help to add variety and precision to your writing.