The Democratic-leaning pollster Public Policy Polling went into Eric Cantor's district yesterday, on the eve of Cantor's surprise defeat at the hands of challenger Dave Brat. They found that Cantor himself was unpopular among registered voters: 63 percent of all registered voters disapproved of his job performance, and his net approval among registered Republicans was -6 percentage points.
But the poll also described an immigration reform proposal similar to the bill that passed the Senate last year, and asked voters whether they supported it. The results might surprise you. From Politico's writeup of the poll:
About 72 percent of registered voters in Cantor's district polled on Tuesday said they either "strongly" or "somewhat" support immigration reform that would secure the borders, block employers from hiring those here illegally, and allow undocumented residents without criminal backgrounds to gain legal status - three key tenets of an overhaul[...]
Looking just at Republicans in Cantor's district, the poll found that 70 percent of GOP registered voters would support such a plan, while 27 percent would oppose.
Obviously, because both the pollster and the group commissioning the poll (the liberal advocacy group Americans United for Change) are left-leaning, it's worth taking that poll result with a grain of salt. (The poll also said that immigrants would receive "legal status" rather than "citizenship.")
Most importantly, Public Policy Polling was talking to all registered voters. The voters who booted Cantor out yesterday were the ones motivated enough to show up for a Republican primary — that's definitely a self-selecting group, and one likely to have stronger (and more conservative) feelings than other voters.
But there are actually older polls of Republican voters that say the same thing: when conservatives are asked about a proposal that looks like the "Gang of Eight" immigration bill, but doesn't include the word "amnesty," they're broadly supportive. An analysis of polling by FiveThirtyEight last year showed that, as long as the poll question specified that immigrants would have to fill certain requirements (like paying back taxes or passing a background check), an average of 72 percent of registered Republicans supported the plan.