“When you talk about self-deportation, you’re not talking about an abstract immigrant,” Segura said. “You’re talking about someone that the respondent knows, cares for and may in fact be related to.”

The Latino Decisions poll also found that 31 percent of Latino voters would be more likely to vote Republican if the party “took a leadership role” on immigration.

Senate Majority Leader Harry Reid, D-Nev, pledged Wednesday to introduce an immigration package next year. He told Republicans they’d reject it at their own peril.

A coalition of Latino groups, including the United We Dream youth network, hopes to work with Reid and House Speaker John Boehner, R-Ohio, to find a solution.

They already have begun strategy sessions. One conversation that’s happening behind closed doors is whether it’s better to push for a larger, comprehensive immigration package or start with a more targeted DREAM Act, which would benefit the most sympathetic illegal immigrants but not necessarily their parents.

Whatever is introduced, the legislation probably would need to be broken into parts to get conservatives on board, said Michael Franc, the vice president of government studies at the Heritage Foundation, a conservative Washington research center.

Republicans are more willing to talk, but the conversation needs to focus more on highlighting economic-growth potential, attracting highly skilled labor and not perpetuating the welfare state, Franc said.

Former Florida Gov. Jeb Bush has made the case in the starkest terms. He told New York Magazine last month that he expects the Democratic presidential candidate to win Texas in 2016, which hasn’t happened since 1980. Some 26 percent of Texas’ eligible voters are Hispanic, the second-largest Hispanic eligible-voter share nationally, according to the Pew Hispanic Center.

“It’s a math question,” Bush said. “Four years from now, Texas is going to be a so-called blue state. Imagine Texas as a blue state – how hard it would be to carry the presidency or gain control of the Senate.”

Republicans leaders more likely will be the ones who trigger change within the party. The chorus continues to grow, from Rubio to Bush to Grover Norquist, the founder of Americans for Tax Reform.

Frank Sharry, the executive director of America’s Voice, which advocates for rights for undocumented immigrants, said he expected Republican senators such as McCain and South Carolina’s Lindsey Graham, who previously supported comprehensive changes in immigration policy, also to play a role.

“This is not rocket science,” Sharry said. “If they want to win the White House again, they need to get immigration off the table.”