Mr. Romney had been courting Cardinal Dolan since April. That month, the two had a private meeting, previously undisclosed, at the chancery in New York, across from St. Patrick’s Cathedral, said Peter G. Flaherty, a senior adviser to the Romney campaign who is Catholic and who served as Mr. Romney’s liaison to the religious community when he was governor of Massachusetts.

“We’re going to have outreach to Catholics in a coordinated, organized effort — state by state, diocese by diocese, parish by parish and pew by pew,” Mr. Flaherty said in an interview…

“Since 1972, the candidate who has won the Catholic vote has won the popular vote as well,” said Robert P. Jones, chief executive and founder of the Public Religion Research Institute in Washington. “The Catholic vote does tend to be on the side of the winning candidate. It’s the quintessential religious swing group.”

But Mr. Jones said he did not see evidence of major shifts among Catholic voters this time — not among white Catholics, who tend to favor the Republican, and not among Latino Catholics, who tend to favor the Democrat.