Proponents of immigration reform have sought to raise the alarm about the trajectory of the Republican race. FWD.us, an organization that supports immigration reform, and is funded in part by the Facebook co-founder Mark Zuckerberg, cited private polling to warn late last fall that support for large-scale deportation would be “disastrous” in a general election.

“Any candidate who in the general supports mass deportation risks negatively impacting the brand of his or her party for generations to come,” the group said in a memo circulated to allies in both parties.

In South Carolina, support for bipartisan immigration reform has not always proved politically toxic. Senator Lindsey Graham, a Republican who supports Mr. Bush, won re-election easily in 2014 after backing the same immigration compromise Mr. Rubio supported.

But voters here are plainly looking to the candidates for reassurance.