Roy Beck of Numbers USA, which pushes for a more restrictive immigration policy, said expanding deportation relief could also fail. "It looks radical," he said of the notion of sharply limiting removals.

Americans must keep pressure on Congress to stop lawmakers from ramming through an immigration reform bill that includes amnesty, says Roy Beck, executive director of NumbersUSA.

The group Numbers USA, which calls for limits on immigration, also grew concerned that Mr. McCarthy might support legislation it considered to be amnesty for people who had illegally entered the country. It directed its members to phone and message Mr. McCarthy's offices, and to drop by his office with material making their case, said Roy Beck, the group's CEO. "We did react on McCarthy, because he's truly in play," he said.

Polling data is mixed on that score. On one hand, data from the right leaning "immigration reduction" organization, Numbers USA, suggests that 46% of Republican evangelicals say they would deport most undocumented immigrants. Another 32% say they would deport some, but "ensure no jobs or assistance for the rest."