Some of the nation’s most influential evangelical groups urged a solution to illegal immigration on Tuesday that defies the harsh rhetoric of the Republican primary race, which continues to undermine Mitt Romney’s appeal to Hispanic voters.

The call by the groups represents a recognition that in one bedrock element of the conservative movement — evangelical Christians — the demography of their followers is changing, becoming more Hispanic, and that Republican leaders risk being out of step with their hawkish talk of border fences and immigration crackdowns like those in Arizona.

Tom Minnery, the senior vice president of policy for one evangelical group, Focus on the Family, said many of the 11 million to 12 million illegal immigrants should be free to “come out of the shadows” and “begin the process of restitution” leading to attaining legal residency.

Mr. Minnery spoke at a Capitol Hill news conference called to announce that more than 150 Christian evangelical leaders, including from the Southern Baptist Convention and the National Association of Evangelicals, were endorsing an overhaul of immigration policy.