on (one's) heels
1. Following close behind someone, often in an annoying way. I'm a preschool teacher, so I've had toddlers on my heels all day. The interior designer can't get any work done with your puppy on her heels!
2. Close to overtaking a fellow competitor. The leading candidate should be concerned about the underdog on his heels. If she does poorly on this exam, she may lose the title of valedictorian to one of the students on her heels.
*on someone's heels
Fig. following someone very closely; following very closely at someone's heels. (*Typically: hard ~; hot ~; right ~. See also on someone's tail.) I ran as fast as I could, but the dog was still hard on my heels. Here comes Sally, and John is hot on her heels.