CHICAGO—More than a century ago, this city reversed the flow of its eponymous river, connecting the Great Lakes with the Gulf of Mexico and defining itself as the can-do capital of the American heartland.

Today, that engineering feat is coming under growing scrutiny, as scientists and politicians intensify their battle against a voracious flying fish that has been traveling up the Mississippi for 20 years. Amid signs that Asian carp have breached the last defensive barrier, calls are mounting for a massive do-over.

"We...