Most Republicans would want to answer that question only after electronic verification was fully implemented. Most Democrats would want to put off the question until there was a path for “earned legalization.” So while both parties accept the value of eligibility screening in principle, the logjam over when and how makes progress unlikely.

As two political veterans who will never run for office again, we have a suggestion: Congress should mandate E-Verify, but pair it with a process for gradually legalizing immigrants, on a case-by-case basis. In short, law-abiding unauthorized immigrants who are already working would, under this plan, be able to come out of the shadows — but without a blanket legalization for all 11 million.

That would fit both the Republican goal of step-by-step reform and the Democratic goal of providing permanent legal status to unauthorized immigrants who satisfy certain criteria (passing a criminal-background check, significant length of time in the country, connections with American relatives). This would put aside, for now, other issues, like visas for high-tech workers, the system of family reunification and the long-term status of the children known as dreamers.