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Frank Sharry, executive director of America’s Voice, an immigration reform advocacy group, said the renewed focus on the issue and Mr. Cantor’s remarks reflect a new openness among Republicans to solving the problem and looking at illegal immigrants as “human beings rather than criminals.”

“You see a movement in tone and more openness to common-sense policy, but the moment of truth will be whether a sufficient number of Republicans in the House will vote for reforms that put illegal immigrations on a path toward citizenship,” Mr. Sharry said. “If they do, they will put this issue behind them and it will help the GOP regain their competitiveness with Latino and immigrant voters. If they don’t, they will be digging an even deeper hole for the party with the fast-growing group of voters.”

Roy Beck, executive direction of NumbersUSA, a group that pushes for stricter immigration limits, said the postelection push for immigration reform is losing steam because the two parties don’t see eye-to-eye on “amnesty” or when the border will be secure enough to give illegal immigrants green cards.

“This is a trial balloon,” Mr. Beck said about Mr. Cantor’s comments. “He wants to send out a signal to particular demographic groups that he has an openness. But … he is not talking about 11 million people. He seems to be defining something that would be offered to less than a million. It is a sign of openness and a sign of real limits.

“It shows Cantor is more reasonable than Democrats at this point. He is showing a willing to do a major compromise, where Democrats are saying no compromise of any kind,” he said.