WASHINGTON – Supporters of bipartisan immigration legislation smoothed the way Friday for likely Senate passage of their handiwork, overcoming last-minute disagreements at the bill's controversial core and tacking on other items certain to build support.

A test vote was set for Monday on the bill, which calls for a military-style surge to increase security at the U.S.-Mexican border. At the same time it sets out a 13-year pathway to citizenship for millions of immigrants living in the United States unlawfully.

Sen. Lisa Murkowski of Alaska became the 11th Republican to announce her support for the legislation in the Democratic-controlled Senate. More were expected to follow, possibly enough to produce 70 votes or more and easily overwhelm its critics.

Some Democrats said a heavy show of support at the end of next week could alter the bill's trajectory in the House, where majority Republicans strongly oppose citizenship for immigrants who came to the country illegally or overstayed their visa. "Hopefully as congressmen look how their senators voted, they will be influenced by it," said Sen. Chuck Schumer, D-N.Y., who has played a major role for Democrats on the issue.

Sen. Jeff Sessions, R-Ala., said the measure's claims of border security were no different than previous assurances. "Time and again, politicians have promised, promised, promised. But they never delivered, delivered, delivered. And that's a fact," he said.

With immigration at the top of President Obama's domestic agenda, White House spokesman Jay Carney labeled the Senate agreement a breakthrough.