hard/hot on somebody's heels
hard on (one's)/the heels
1. Moving very closely behind one. In this usage, a noun or pronoun is used between "on" and "heels." Since the dog was hard on my heels, I was very relieved to reach my house and run inside.
2. Close to defeating an opponent. In this usage, a noun or pronoun is used between "on" and "heels." We need to tighten up on defense because these guys are hard on our heels right now.
3. Happening very soon after something else. In this usage, the phrase is written as "hard on the heels of (something)." I feel like I've been sick for months, since I got bronchitis hard on the heels of a lengthy cold.
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.