It is named for a dominating Washington figure and located on the city’s most prestigious avenue, but the J. Edgar Hoover FBI Building is unloved.

Condominiums have sprouted nearby in recent years, and neighbors glower at the concrete planters that surround the government building. City officials consider the Brutalist structure on Pennsylvania Avenue a drab hulk in the midst of an otherwise colorful and rising downtown. Even the FBI says the building no longer meets its needs.

And yet, the real estate where the complex sits, on nearly two city blocks in the heart of downtown, could not be more valuable in these days of federal budget turmoil.

So with pressure building to find savings in the federal real estate portfolio, the General Services Administration on Monday proposed handing the Hoover building to private developers in exchange for building the FBI a new headquarters campus elsewhere in the region.

“Our thought is we can trade the current facility and its location for something new and more efficient that fits the current requirements of the FBI,” GSA Acting Administrator Dan Tangherlini said.

President Obama has directed the GSA to sell or use properties that are vacant or under-utilized, opening up the possibility that big chunks of block-like federal real estate could be transformed into new offices and city neighborhoods lining Pennsylvania Avenue and the Mall.

Nearly a year ago the GSA chose Donald and Ivanka Trump’s development company to turn the Old Post Office Pavilion into a luxury hotel. The agency is also seeking private-sector ideas for a 22-acre area consisting of five office buildings near the Mall in Southwest Washington.

Called Federal Triangle South, the dull collection of properties could be redeveloped into a neighborhood of new housing, hotels, offices and shops connecting to a Southwest Waterfront pegged for redevelopment beginning next year.

A similar neighborhood could replace the FBI site downtown, on a magnitude eclipsing the $950 million CityCenter project under construction a few blocks north.

“What we want to do is ask the marketplace: What do you think?” Tangherlini said.

The GSA’s search immediately kick-started a competition among local jurisdictions to win a federal campus that would bring as many as 11,000 FBI headquarters jobs. Local members of Congress sounded like real estate brokers as they pitched sites and talked up their jurisdictions.

Del. Eleanor Holmes Norton (D-D.C.) said the FBI needs to stay in Washington to remain in close contact with policymakers; she said there are at least five suitable sites in the city.

“The FBI is the last agency that would want to be out of the District of Columbia,” she said. “It needs proximity to Capitol Hill and the White House on a daily basis. And when I say proximity, I mean rapid proximity.”

But officials in Fairfax and Loudoun counties have also pitched potential sites ever since a Government Accountability Office report last year said relocation would be cheaper than making renovations that could require $1.7 billion over 14 years. Exxon-Mobil’s decision to vacate a campus in Merrifield opens up a possible landing spot.

Prince George’s County leaders have been adamant that the GSA move more federal office jobs to their jurisdiction, and they point to 15 Metro stations with ample available land. Rep. Donna F. Edwards (D-Md.) and others blasted the GSA’s decision last year to keep a 15-year lease for the Department of Health and Human Services in Rockville.

Edwards said she would not advocate for any particular site, but said taxpayers and the region were likely to get the best deal out of an FBI move to Prince George’s, which has possible sites at Greenbelt, Branch Avenue, Landover Mall and elsewhere, according to county officials.

“The reality is that something on the order of 65 percent of our federal workforce in our county are driving somewhere else to go to work,” Edwards said.

Named by Obama to lead the GSA after a conference spending scandal, Tangherlini said he was committed to working with county officials. But he said he hoped the competition would result in more savings for taxpayers. “If jurisdictions are competing, then we at the GSA and federal taxpayers are going to get better deals,” he said.

When the Hoover building was completed in 1974, its concrete walls and canyonlike dry moat were considered top-shelf security measures. But because they fall short of current requirements, the FBI has closed main entries, canceled public tours and surrounded the building with planters. The building accommodates 54 percent of the agency’s headquarters staff; the rest are scattered in offices elsewhere.

“The FBI cannot afford to continue the status quo, from an operational effectiveness or a fiscal stewardship perspective,” said FBI associate deputy director T.J. Harrington in a letter to the GAO last year. “A new consolidated FBI headquarters facility is urgently needed and we view this as one of our highest priorities for the foreseeable future.”

While the FBI voiced its displeasure about the building in recent years, a snazzy new downtown neighborhood grew up around it, one featuring retailers H&M, Anthropologie and shops selling frozen yogurt and gourmet sandwiches.

The apartments and restaurants make the concrete mass of the government building appear more dated than ever. Its sidewalks, devoid of cafes and patrolled by FBI authorities on Segways, seems to belong to a time when downtown D.C. emptied at 5 p.m. Even top historic preservation officials in the city consider the building not worth what would probably be a very public, bitter fight to try to save it.

“It’s hideously ugly,” said LeTesha Dixon, a dog walker who passes the property on weekdays. She said she thinks the downtown area, where more than 56,000 people live, needs more open space. “Part of it should be a public gathering place,” she said.

The Hoover building was constructed at a time when the government needed room for fingerprint records, investigative reports and files, a requirement computers have rendered largely unneeded.

“It was really seen at the onset as a giant filing cabinet,” Tangherlini said.

He said he hoped the sale of the Hoover building, which was completed shortly after the agency became enmeshed in the Watergate scandal, would pay for all or most of a newly constructed FBI campus.

“It was built for a purpose and a use that has really been overwhelmed by time,” he said.