Prediction markets won bragging rights last year when they correctly called the U.S. presidential election for Donald Trump . Now, they’re taking on a trickier challenge: forecasting the winner of next week’s papal conclave.

At Polymarket, a popular crypto-based platform, users have traded more than $12 million worth of contracts that will pay off if their holders make accurate predictions about the next leader of the world’s 1.4 billion Catholics.

Meanwhile, around $4 million worth of similar contracts have changed hands at Kalshi, a rival prediction market. Kalshi—which unlike Polymarket is open to Americans—began taking wagers on the next pontiff just hours after the death of Pope Francis on April 21.

The Roman Catholic Church once threatened people betting on papal conclaves with excommunication, and laws in many jurisdictions forbid wagering on the next pope.

But as prediction markets become an increasingly popular way to forecast elections , the odds on Polymarket and Kalshi will likely be closely followed as cardinals convene in the Vatican on Wednesday for the first conclave in more than a decade. The secretive process unfolds behind closed doors in the 15th-century Sistine Chapel, and a successful final vote is announced with a puff of white smoke from the chapel’s chimney.

The current front-runner is Cardinal Pietro Parolin , the Vatican’s secretary of state, with a 23% chance of winning as of Thursday afternoon, according to Polymarket. In second place is Cardinal Luis Antonio Tagle , a Filipino dubbed the “Asian Francis,” at 19%.

Another favorite among some bettors is Cardinal Peter Turkson of Ghana. The African cardinal’s chances on Polymarket mysteriously surged from 7% last week to 17% on Thursday, putting him in third place. The same three cardinals are top contenders at Kalshi, as well as at U.K. bookmakers that allow betting on the next pope.

Betting odds have a mixed track record of predicting papal elections. During the conclave that followed the 2005 death of Pope John Paul II , Cardinal Joseph Ratzinger rose in the odds to become the favorite, and ended up becoming Pope Benedict XVI. But in 2013, Cardinal Jorge Bergoglio —the future Francis—was a distant underdog, with U.K. bookmakers giving him less than a 4% chance of winning.

Unlike U.S. elections , papal conclaves don’t have opinion polls, public debates or campaigning by leading contenders. That makes it harder for bettors to forecast the outcome, according to Leighton Vaughan Williams , an economics professor at the U.K.’s Nottingham Business School. undefined undefined “Current betting odds, favoring cardinals like Parolin and Tagle, probably reflect general perceptions, media attention and public profile rather than special insider knowledge or highly accurate predictive insight,” Vaughan Williams said.

Catholic teaching says the Holy Spirit helps guide cardinals in their selection of the next pope . For bettors in prediction markets, the thinking isn’t as lofty. Some favor Cardinal Pierbattista Pizzaballa , an Italian who serves as Latin Patriarch of Jerusalem, because they like his name. “Just bet on the pizza dude for pope. Don’t overthink it,” one user posted in a Discord forum for Kalshi users last week.

Josh Weil , a 36-year old alternative healer in Santa Barbara, Calif., placed small bets on Turkson and another African, Cardinal Robert Sarah of Guinea, using Kalshi. He also bet against Tagle, reasoning that the 67-year-old Filipino was too young and too liberal to be chosen pope.

“My investment thesis is that it’s time for a Black, conservative pope,” said Weil. “It’s not really scientific, but I just feel like it’s that kind of cultural moment.”

Betting on papal conclaves has a long history. In 16th century Rome, both commoners and elites took part in a bustling market for such wagers, fueling a rumor mill in which the closed-door machinations of the conclaves often leaked to the public, according to John Hunt , an independent historian who studies the papacy.

Following a series of scandals—including brokers spreading false rumors to influence the market—the authorities cracked down. In 1591, Pope Gregory XIV issued a bull that banned all wagering on papal elections on penalty of excommunication.

The bull remained in effect until a broad reform of church canon law in the early 20th century. But betting on papal conclaves is illegal in Italy today, under a government decree that restricts wagers to an approved list of events, mostly sports-related. The list doesn’t include any religious events. Vatican law, meanwhile, bans gambling in the city-state.

In the U.S., traditional casinos and sportsbook operators are barred from offering bets on papal conclaves under state laws that generally limit wagers to sporting events. Kalshi, which has a federal license from the Commodity Futures Trading Commission, won the right to list contracts tied to elections last year after winning a court battle with regulators.

Write to Alexander Osipovich at alexo@wsj.com