The two sides said they have agreed to three principles. The first is that American workers should have the first crack at all jobs, and the second would provide a new visa that “does not keep all workers in a permanent temporary status, provides labor mobility . . . and that automatically adjusts as the American economy expands and contracts.”

The third principle is a call for a new, quasi-independent federal bureau that would monitor employment statistics and trends to inform Congress about where to set visa caps for foreign workers each year.

A spokesman for Sen. Marco Rubio (R-Fla.), who is helping lead talks in the Senate, praised the union-business negotiations while also criticizing Obama for not endorsing a guest-worker program.

“It’s good the unions have recognized there must be a workable visa program for lesser-skilled workers as we reform our legal immigration system,” Rubio spokesman Alex Conant said.