"Republicans realize the implications of the future of the Republican Party in America if we don't get this issue behind us," he said.
That now falls to Boehner's chamber, where conservatives in his party have complicated his agenda on other subjects -- few with such long-term implications as immigration.
Republicans and Democrats alike were watching Boehner's next move.
"I'm hopeful that we can convince our House colleagues," McCain said. "I believe that Speaker Boehner has a tough job ahead. I admire his leadership. But we've got a lot of work to do."
If it fails, Democrats stood eager to blame Boehner and his party.
"Will he allow a small group, maybe even a majority of his caucus, to control the debate and the future on this issue?" asked Rep. Luis Gutierrez, D-Ill. "If he decides to do that, we will then end in a stalemate and an impasse once again."
But an immigration bill could be trouble for Boehner's rank-and-file members.
"They fear Republican primaries from the right if they vote `yes,'<!p><#148> said Sen. Chuck Schumer, D-N.Y.
But Democrats were unlikely to yield on their principles, warned Schumer, who helped write the Senate bill.
"No Democrat will vote for any bill without a path to citizenship," Schumer said.
Democrats in the House, though, urged their GOP colleagues to keep working toward an immigration overhaul despite its political risks and fierce intraparty infighting.
"I know it's going to be a tough slog in the House. I've seen what's been going on in that place," said Rep. Donna Edwards, D-Md. "But we can get this done and have to get it done for the country."
Goodlatte and Gutierrez spoke to CNN's "State of the Union." Pelosi was interviewed on NBC's "Meet the Press." Gowdy, McCain and Schumer were on "Fox News Sunday." Edwards appeared on ABC's "This Week."