Shortly before the 2002 midterm elections, Richard A. Gephardt, who was then the House minority leader, infuriated his fellow Democrats when he announced he would help President George W. Bush win authorization for the Iraq war.

“It’s really the most difficult thing you have to ever do — take on a large part of your caucus on something they feel strongly about,” Mr. Gephardt, who represented Missouri, recalled in a recent interview. It is precisely what Speaker John A. Boehner would have to do with his fellow Republicans to move immigration legislation toward a landmark compromise with the Democratic-led Senate and President Obama. And in the unlikely event he tries, Mr. Boehner would face an even more difficult task than Mr. Gephardt.

But treacherous as it would be, the prospect offers Mr. Boehner tantalizing benefits. It could return him and the House to the legislative problem-solving he has long valued but has rarely been able to accomplish as speaker.

It could engrave on his record the adaptation of American law to historical, cultural, demographic and economic changes. It could help his party stem its alienation from the swelling Hispanic electorate and revitalize its ability to win presidential elections. Still, the reasons not to try have hardened into a familiar Washington political consensus.