The phrase "rival clubs" refers to two or more clubs or organizations that compete against each other and have a strong sense of competition or rivalry between them. Full definition
Both players are currently set to be out of contract in the summer, with various rumours of rival clubs interested in snapping the pair up. (caughtoffside.com)
As the name implies, you're part of a racing club competing against other rival clubs. (teamvvv.com)
You keep seeing rival clubs do well, winning trophies and challenging for the titles you wish your club can compete for. (arsenalfcblog.com)