There are several synonyms for the phrase "giving off odor." Some options might include emitting a scent, exuding a smell, or producing an aroma. Other possibilities might include releasing a stench, emanating a fragrance, or sending forth an odor. Depending on the context, there may be more specific terms that could be used as well. For example, if the odor is particularly unpleasant or foul, you might describe it as reeking, putrid, or nauseating. If the smell is more pleasant, you might use terms like fragrant, sweet-smelling, or aromatic. Overall, there are many different ways to describe the act of giving off an odor, depending on the specific connotations and nuances you wish to convey.