PAOLI, Pa. — Since the election of Ronald Reagan, white Roman Catholics have flocked to Republican nominees for a raft of reasons, including their stances on social issues like abortion and same-sex marriage.

But this year, something seems different.

“Trump is the exception to the rule,” Carol Robinson, 67, said as she left an afternoon prayer meeting in this Philadelphia suburb with other enthusiastic supporters of Hillary Clinton. “He’s a loose cannon.”

Roman Catholics are the country’s second-largest religious group after evangelical Protestants, and they are as diverse as the country itself, with young liberals, cultural conservatives and, increasingly, Democratic-leaning Hispanics.

But now, the Clinton campaign senses a rare opportunity to block Mr. Trump’s narrow path to victory by making inroads with a core part of the church: white Catholics, a prized group of voters that has defied predictions this year.