The construction of a massive new headquarters for the Department of Homeland Security, already more than $1.5 billion over budget and 11 years behind schedule, may face even more overruns and missed deadlines because the government has still not developed reliable cost estimates, federal auditors said Friday.

In a report to Congress, the U.S. Government Accountability Office said DHS and the General Services Administration should consider alternatives to the plan to locate the headquarters complex on the grounds of St. Elizabeths, a onetime insane asylum with a panoramic view of the District. The project, billed as critical for national security and the revitalization of Southeast Washington, calls for renovating dozens of historic buildings, but it has been starved of funds and only one new structure has opened.

“Creating reliable cost and schedule estimates for the headquarters consolidation project should be an integral part of DHS and GSA efforts to reassess the project,’’ the report said. “Without this information… the project risks potential cost overruns, missed deadlines, and performance shortfalls.’’

Republicans seized on the 77-page document at a House homeland security subcommittee hearing Friday, saying the project should be reevaluated at a time of fiscal austerity and its funding steered to other DHS priorities such as border security.

“I think Americans would love to see us redirect some of these resources,’’ said  Rep. Jeff Duncan (R-S.C.), chairman of the homeland security committee’s oversight subcommittee. “We need to prioritize where we spend our taxpayer dollars.’’

DHS and GSA officials defended the project, urging the subcommittee to fully fund it and saying it would save money in the long term because DHS could get out of expensive leases it is paying to house pieces of the department around the Washington are. “When we receive  consistent funding, we can deliver on time and on budget,’’ said Chris Cummiskey, DHS’s acting undersecretary for management.

Senate Democrats on Friday offered strong support of the St. Elizabeths plan, which was developed by the George W. Bush administration soon after DHS was created in the wake of the Sept. 11, 2001, attacks.

In what amounted to a dueling report, Democrats on the Senate Homeland Security and Governmental Affairs Committee released a document calling the headquarters consolidation a top priority.

Their document emphasized that DHS operates out of more than 50 facilities around the Washington area, making it difficult to fulfill the agency’s mission of protecting the homeland from terrorism and responding to natural disasters. DHS has long cited this unwieldy structure to support St. Elizabeths, with officials saying logistical problems slowed the government response to Hurricane Katrina in 2005 and to various terrorist plots.

“The Department’s ability to accomplish its mission, promote employee productivity, and communicate within its components remains difficult. We need to fix this,’’ said the Democratic report, prepared by the staff of Sen. Thomas Carper (D-Del.), the committee chairman. “Consolidating the Department at St. Elizabeths will improve DHS’s ability to carry out its mission.’’

In one memorable example, the Democrats’ report said current DHS facilities are so subpar that a recent “issue with cleanliness and a rat-infestation” temporarily required closing the cafeteria at the current headquarters complex on Nebraska Avenue in the District. DHS officials have long complained that the headquarters is one-quarter of the size they need, with a small operations center plagued by limited infrastructure.

The back and forth shows that consensus over the headquarters project remains elusive a decade after it was first conceptualized. The entire complex was to be finished as early as this year, at a cost of less than $3 billion, according to the initial plan. That budget has ballooned to $4.5 billion, with completion pushed back to 2026. With the exception of a Coast Guard building that opened last year, the grounds remain entirely undeveloped, with the occasional deer grazing amid the vacant Gothic Revival-style structures.

Afflicted by partisan brawling for years, St. Elizabeths — the capital region’s largest planned construction project since the Pentagon — has also received only lackluster support from the Obama administration, according to budget documents and interviews with current and former federal officials.

The project’s cost has been controversial from the start, and Friday’s GAO report says costs estimates have been plagued by “numerous deficencies” that render them unreliable. For example, the report said, a revised 2013 estimate did not include the cost of operations and maintenance. Even DHS and GAO, it said, admitted it was “difficult or impossible” to specify the project’s final price tag because it is so complex.

GAO also faulted federal officials for not updating the St. Elizabeths master plan since it was approved in 2009 and said the plan no longer takes into account changes in how people work such as telework.

In a written response to the report, Jim H. Crumpacker, a DHS official who oversees dealings with GAO, said the department is “disappointed with much of the context and perspective’’ in the report and said the shortfalls in funding from Congress “cannot be overemphasized.’’ But he said the department will work with GSA to respond to GAO’s concerns.

GSA administrator Dan Tangherlini also said in his response that the two agencies would work together to “remedy the concerns raised by GAO” and said the funding shortfall has created “a serious challenge” for the headquarters project.