CARACAS, Venezuela — One of Venezuela’s most prominent opposition leaders threatened to break from the broad alliance of opposition parties on Tuesday, pushing the coalition to the brink of fracture and further deepening the nation’s political crisis.

Henrique Capriles, a two-time presidential candidate who narrowly lost to President Nicolás Maduro in 2013, coupled his threat with a full-throated demand for an overhaul of the alliance.

“This is like when you go to the doctor and they detect a tumor,” Mr. Capriles said in a broadcast on the Periscope live-streaming application. “It’s time to remove the tumor and cure the patient, Venezuela.”

Mr. Capriles’s announcement highlighted the deepest public rift in the alliance since its founding about a decade ago. It came a day after the governors-elect from one of the coalition’s most influential parties took the oath of office before the Constituent Assembly, an all-powerful body engineered by Mr. Maduro, infuriating many in the opposition.