In the days surrounding Pope Francis’ death, Cardinal Pietro Parolin was everywhere.

The No. 2 figure in the Vatican, he visited Francis in the hospital, and then helped seal the papal apartments after the pope died. He welcomed cardinals he knew from around the world to the pope’s funeral, talking to former President Joseph R. Biden Jr., President Volodymyr Zelensky of Ukraine and former Prime Minister Matteo Renzi of Italy. And leading up to the conclave to pick the next pope, he celebrated an outdoor Mass for tens of thousands of faithful on the steps of St. Peter’s Basilica.

It seems that everyone knows Cardinal Parolin, the Vatican secretary of state, who will preside over the papal election and who has emerged as the leading compromise candidate before a conclave in which many of the more than 130 cardinals do not know one another.

“The only candidate that for now has emerged with a certain insistence is Parolin,” said Andrea Riccardi, the founder of the Sant’Egidio Community, a Catholic group close to Francis. “He represents a continuity,” added Mr. Riccardi, who is close to several cardinals who are considered papal contenders. “He has said, ‘Virtue stands in the middle.’”

A quiet, plodding Italian with a famously inscrutable poker face, Cardinal Parolin is deeply cautious. But at a time of global upheaval, that is not necessarily a disqualifier. Even his backers grant that he lacks Francis’ charisma and global symbolism — but as the leader of the Vatican machinery for the past decade, he enacted Francis’ vision.

Cardinals have talked about Cardinal Parolin as someone who could have a steady, bureaucratic hand on the church’s wheel. And at 70, he could appeal to cardinals who do not want to be stuck too long with the winner.