The House will vote Thursday on an appropriations measure that would cut funding for construction of the Department of Homeland Security headquarters at the former St. Elizabeths Hospital site in Southeast D.C.

The bill, which is expected to pass but still must be considered by the Senate, covers Homeland Security spending for fiscal 2012. It does not include the $160 million requested by DHS to continue the St. Elizabeths project, which will consolidate dozens of agencies with thousands of workers currently scattered at different locations onto one sprawling campus. The project will also create thousands of construction jobs.

The cut is part of a broader effort by House Republicans to save money across the government. And the Appropriations Committee, which authored the bill, has complained about the way the venture has been managed, saying “both costs and schedule of the current project are matters of concern” to panel members.

The cut has drawn the ire of local officials as well as of the Obama administration.

DHS has enough money already to complete a headquarters building for the Coast Guard, but beyond that, the White House said in a Statement of Administration Policy issued this week, the “bill would delay the consolidation of the Department of Homeland Security headquarters by at least two years, resulting in higher lease costs and will mean the loss of construction efficiencies and increased future construction costs.”

Del. Eleanor Holmes Norton (D-D.C.) made similar complaints. On Wednesday, during debate on the measure, Norton offered an amendment to restore funding for the DHS project, but she withdrew it before it could come to a vote because it likely would have been rejected.

“If Republicans are serious about reducing government spending, they will support my amendment, which will save taxpayers from the increased costs, both now and in the long-term, of consolidating DHS into one facility and eliminating the leasing of space at several dozen facilities,” Norton said in a statement before she offered the amendent Wednesday.

A Norton spokeswoman said she is focused on making sure the money is included when the Senate considers the bill. That chamber, which is controlled by Democrats, is unlikely to support the same cut advocated by House Republicans.