In the budget, released Monday, Mr. Trump says he backs the current mostly voluntary version of E-Verify.

E-Verify, run by Homeland Security, is an electronic system available to employers to run their new hires' information. They are told, usually instantaneously, whether the person is work-authorized.

Currently it's voluntary for most companies, though federal contractors are supposed to use it, and some states have passed laws mandating use as well.

Though not infallible, the system is considered the best way to prevent illegal immigrants from taking jobs, and has long been the Holy Grail for security advocates and those pushing for stricter immigration limits.

The Trump budget retreat was dismaying to them.