Proposals to expand visas for high-tech workers, streamline a guest-worker program for the nation’s farms, establish a mandatory E-verify system for businesses and create an exit-visa registry to rein in overstays could all pass both chambers with bipartisan support, the GOP reform advocates said.

“It’s a losing issue for the Democrats if the Republicans just take this step-by-step,” Norquist said.

Obama and the Democrats have largely opposed that piecemeal approach for fear that passing the more popular elements of immigration reform would make it all but impossible to move more contentious provisions, particularly the creation of a pathway to citizenship for illegal immigrants.

But influential Republicans say the president would be hard-pressed to veto legislation that hit his desk with Democrats on board.

“Republicans can pass these in the House and Senate and the president’s going to have to veto or sign it — put up or shut up,” Norquist said. “He can’t play this game, ‘How come you guys won’t do X?’ We just did X, buster.”