PHOENIX — A push to assuage opposition to the bipartisan immigration bill before Congress by devoting more money and muscle to the task of securing the border with Mexico has yielded at least one unforeseen consequence: It weakened support for the bill among some pro-immigrant groups that had been its most reliable backers.

Advocates have staged protests in several cities this week denouncing a plan endorsed by the Senate to inject $40 billion in enforcement measures over the next decade, including 18,000 more Border Patrol agents and 700 more miles of the hulking steel fence that demarcates the countries.

Leaders of Presente.org, the nation’s largest online Latino advocacy organization, took the step of opposing the broader immigration bill altogether, saying in a statement they could not “in good conscience” stand by it if it is also “guaranteed to increase death and destruction through increased militarization of the border.”

Other advocates are considering the same path as they increasingly shift their criticism to the Democrats. In closed-door meetings, many have accused Democrats of giving up on a balanced compromise over immigration reform just to move the bill forward.