Sure, you can do that. But aside from the ethics of it, there's always a chance that they won't counteroffer and just say "Maybe you should go with that higher

Sure, you can do that. But aside from the ethics of it, there's always a chance that they won't counteroffer and just say "Maybe you should go with that higher offer." You really don't want to bring up a "competing offer" unless you're prepared to take the counteroffer/just walk away.