hot on (one's) heels
1. Following close behind someone. Look, the cops are hot on their heels! They better pull over fast.
2. Close to overtaking a competitor. The leading candidate should be concerned. The underdog is hot on his heels. If she does poorly on this exam, she may lose the title of valedictorian to one of the students who are hot on her heels.
hard/hot on somebody’s ˈheels
following somebody closely because you want to catch them: Jane has the most points at the moment, but the other competitors are hot on her heels.The police are hard on his heels.