The immigration provisions are included in a bill to fund the Department of Homeland Security, a sprawling agency that secures U.S. borders, airports and coastal waters.

Senate Republican leaders warned last week it would be difficult to get the 60 votes needed to overcome Democratic procedural hurdles. Republicans now control the Senate with a 54-vote majority.

Representative Raul Labrador, a Tea Party favorite from Idaho, said at a news conference with other conservatives that it was “high time that (Senators) Ted Cruz and Mike Lee and others decide that they are going to start fighting in the Senate using the Senate procedural rules, and not just looking at the House as a place where the fights are going to happen.”