hard on someone's heels or hot on someone's heels
doing nearly as well as someone else in a competitive situation, and likely soon to be doing better than them
The next generation of British athletes is pressing hard on the heels of today's champions.
close behind someone, often chasing them
Two RAF planes appeared overhead, but the pursuers were hard on the paras' heels.
Easy Learning Idioms Dictionary. Copyright © HarperCollins Publishers