Mr. Schumer, who wants to be the next Senate majority leader, had worked on the 2007 push under his mentor and legendary deal maker, the late Ted Kennedy. “I called Schumer and said ‘Let’s get the band back together,’ ” says Mr. Graham. The Republican invokes the economic and moral arguments for immigration but emphasizes the political one. If the GOP brand with minorities doesn’t improve, the party won’t get out of an “electoral death spiral.” “After the 2012 election,” he says, “my worst fears were being realized.”

Of the sponsors, only Mr. Graham faces an imminent election. He says Mr. McCain was worried the reform push would invite a primary challenge, but “I said, ‘Listen, name a time better than now.'” He cites polls that most voters are with him. “The bottom line is [that] from a political point of view, it’s been hurtful to me overall. But I think I’ve turned a corner, because the people I’ve lost, I’ve lost,” he says. A fifth of Republican primary voters oppose the legal pathway to citizenship.

“Schumer’s been incredible,” Mr. Graham says. “He’s a worthy successor to Ted Kennedy, and that’s saying a lot.” Marco Rubio, who joined the Gang of Eight in January, “has been a game-changer who’s been terrific on our side.” Dick Durbin, Mr. Graham says, is “tough but practical, always fighting for [organized] labor’s interest.”