VATICAN CITY—The prospect of Pope Francis agreeing to let China’s Communist government control his selection of the country’s Catholic bishops has stirred one of the biggest controversies of his reign, with critics denouncing the plan as a betrayal and a capitulation.

“I would make a cartoon showing the pope kneeling and offering the keys of the kingdom of heaven and saying, ‘Now, please recognize me as pope,’” Cardinal Joseph Zen, retired bishop of Hong Kong, the most prominent critic of the prospective deal, told an interviewer...