“It seems something’s broken about the system,” Mr. Crenshaw said.

The shift among evangelical Christians could have a powerful effect on the fight in Washington, as Republican lawmakers, including many who have opposed any amnesty for illegal immigrants, look to see how much they can support measures to bring those immigrants into the legal system without alienating conservative voters.

Evangelical leaders, seeing the opportunity to expand their influence on a social issue beyond abortion and same-sex marriage, have broadly united this year behind a path to citizenship for immigrants in the country illegally. They are conducting an ambitious push to sway Congress, including ad campaigns on Christian radio stations in five states, meetings with lawmakers and a challenge to churchgoers to pray every day for 40 days using Bible passages that speak of welcoming the stranger.

On Wednesday, evangelical pastors will converge on Washington for a day of prayer and lobbying on Capitol Hill.

Guiding the campaign is a coalition called the Evangelical Immigration Table, which includes the top pastors of more than two dozen evangelical denominations and at least 20 heads of Christian colleges and seminaries. “It is very remarkable the degree to which there is consensus,” said Galen Carey, the vice president of government relations for the National Association of Evangelicals, an umbrella group for the churches. No prominent pastor has spoken out against the immigration effort, although some pastors of the largest churches have remained silent.

Accord has been less broad among the faithful. In a poll released in March by the Public Religion Research Institute and the Brookings Institution, white evangelical Protestants were the least likely of the religious groups surveyed to support a path to citizenship for immigrants here illegally, with 56 percent of them favoring that approach. Among Hispanic Catholics, the group expressing the most support, 74 percent said they would allow those immigrants to become citizens. Only 41 percent of white evangelicals who identify with the Tea Party supported a path to citizenship, according to the survey.