"At someone's heels" means to follow closely or pursue persistently, and there are several synonyms for this expression. One could say that someone is "on their tail," "hot on their heels," "in pursuit," "chasing after," "dogging," "hounding," or "trailing closely" behind them. These phrases could be used in various contexts, such as in a race or a chase scene in a movie. Additionally, one could use similar expressions to describe a feeling of being closely monitored or watched, such as being "under someone's watchful eye" or "being closely observed." These synonyms add variety to one's vocabulary and can help to make their writing or speech more engaging and effective.