Graham, something of a Democrat-whisperer for conservatives, may understand better than anyone the strength of the other party’s paranoia. “Here’s the problem for our Democratic colleagues,” he said Tuesday. “If you say 90 percent operational control of the border, it wouldn’t be hard to envision a Republican-controlled wing of the Congress where they undercut the ability to get to 90 percent through lack of funding.”

On all but a few key points, Cornyn’s amendment is in line with the border provisions in the underlying bill. The base bill includes the same border-apprehension and surveillance benchmarks. Cornyn’s amendment adds to that by putting citizenship eligibility on the line. The base bill includes an entry-exit system to keep track of all foreigners who enter and leave the country. Cornyn’s amendment adds to that by requiring fingerprints at the most used airports and seaports.

Democrats, knowing many Republicans whose support they will need on final passage approve of the border-trigger concept, say they are willing to reshape Cornyn’s proposal such that it will answer their concerns. “We’re trying. We haven’t given up on it because we’d like to have John’s support,” said Sen. Dick Durbin of Illinois, the top-ranked Senate Democrat in the “Gang of Eight” that drafted the immigration bill.