MOSCOW—The hard hats are on. The bulldozers are purring. Soon enough, the maze of concrete and metal inside Moscow’s hollowed-out Luzhniki Stadium will transform into a soccer palace rebuilt to FIFA’s every whim.

Construction for the 2018 World Cup is gaining momentum in Moscow and cities across Russia, even as soccer’s global governing body faces corruption probes in the U.S. and Switzerland, marking one of the deepest scandals in the sport’s history.

Some...