Still, Mr. Obama had not lost all his friends in the church. As the president’s relations with Catholic leaders reached their nadir, Father Stenzel, Mr. Obama’s old smoke-break friend, visited the White House. As they walked into the Oval Office, Mr. Obama joked to his staff that the priest had given him his first office in Chicago. Father Stenzel reminded him that his old surroundings were far humbler: “The office I gave you had two rows of glass-block windows!”

Pope Francis’ Impression

Mr. Obama’s parish days seemed far behind him when he won re-election in 2012 with a slimmer margin of Catholic votes. Not only did Catholic conservatives view him as a secularist forcing them to pay for contraceptives, but some of his old allies in the church’s left wing criticized his use of drones and lack of emphasis on the poor.

But the election of Pope Francis last March seemed to breathe new life into the Catholic Church and, potentially, into the relationship between Mr. Obama and the institution that gave him his start. While far from an ideological progressive, Francis does sometimes appear cloaked in Cardinal Bernardin’s “seamless garment.” His de-emphasis of issues like abortion and same-sex marriage and his championing of the poor and vulnerable — articulated in his mission statement, “The Joy of the Gospel” — have impressed a second-term president who argues that income inequality undermines human dignity.

“Whether you call that the ‘seamless garment’ or ‘the joy of the Gospel’ or what, I’ve said to the president I consider that a pretty Catholic way of looking at the world,” said Denis McDonough, the White House chief of staff, who is Roman Catholic. Mr. McDonough added that the community-organizer-turned-president had expressed admiration to him about “how important it is for the Holy Father to be so in the community.”

Last month, Catholic activists made their case for social justice on Capitol Hill. Afterward, relaxing over beers and a buffet in the Russell Senate Office Building, they discussed whether Cardinal George, who is retiring as archbishop of Chicago, would be replaced by Archbishop Gregory, who helped secure Mr. Obama’s church grant application in the 1980s. Among them was Mr. Lyke, the man who had received coaching from Mr. Obama years earlier in the basement of Holy Name Cathedral. He characterized Francis and Mr. Obama as a match made in heaven.

Mr. Lyke’s view is not universal. Vatican officials have made clear Mr. Obama will not get special treatment, and leaders of the Catholic Campaign for Human Development, also gathered in the Russell Building, saw the coming papal audience as a chance for Mr. Obama to return to the church’s social justice values, not the other way around.

Dylan Corbett, one of the Campaign for Human Development leaders, said the president was “welcome to the conversation” that the pope was driving about income inequality and poverty. He added with a grin, “We’re happy to have him back, actually.”