The fast-growing CoStar Group has a chic downtown corporate headquarters and is now aiming to expand into one of the city’s most historic buildings, the Franklin School.

Andrew Florance, chief executive of the commercial real estate data firm, said his company has nearly outgrown its 168,000-square-foot L Street headquarters. He is targeting the red brick former school building overlooking Franklin Square for more space.

The son of a Cleveland Park architect, Florance founded CoStar in his parents’ basement in 1987 and turned it into a data juggernaut that had revenue of $349.9 million in 2012. He said he has known the Franklin School building practically his entire life and lamented seeing it fall into disrepair in recent years. The building, which had been used as a homeless shelter, now sits vacant.

“Franklin School is one of the city’s most important architectural gems,” he said. “It is a building of very significant historic importance to the city. And it is rotting.”

CoStar is among the companies and organizations that have expressed interest in the building since Mayor Vincent C. Gray (D) began seeking partners to redevelop it. Responses to the search were due Friday. The office of Victor L. Hoskins, Gray’s deputy mayor for planning and economic development, has not yet released information about what companies responded.

Designed by architect Adolf Cluss, Franklin School has been used for educational purposes throughout almost its entire history. In 1876, it was the site of experiments by telephone inventor Alexander Graham Bell, and four years later, it became the city’s first public high school. It was the headquarters for the school system from 1928 to 1968.

Recent years have been far less kind. The District government turned it into a homeless shelter in 2002. Some 400 men at a time stayed there over the next decade, but the building suffered damage from leaks, rodent infestation and mold. Mayor Adrian Fenty closed the shelter under protest in 2011. Today, paint peels from the walls and ceiling, and wide holes and gaps are visible in the plaster.

Florance is proposing to rehabilitate Franklin into open offices for CoStar’s technology training and operations. He said the company already has about 250 software developers and technology positions and that it needs to train more. A funky downtown office fashioned into a building listed on the D.C. Inventory of Historic Sites might make just the place, he said.

“I could see that some of that highly creative space could turn off a lot of tenants. It’s something that would be far more attractive to a company like us. All of CoStar’s offices globally have an open floor plan,” he said.

CoStar isn’t likely to be the building’s only suitor; 139 developers, architects and other professionals toured the building with D.C. officials after Gray began his search.

But for all its grandeur, Franklin School may struggle as a commercial enterprise. Although it is listed at 51,000 square feet, District officials acknowledge that because of the building’s configuration, only 33,000 to 38,000 square feet are likely to be usable. It is so important historically that its exterior, its stairwell, its roof trusses and even the remaining frescos — or plaster murals — on interior walls must be preserved. There is no underground parking.

For those reasons and others, previous efforts to develop the building have fallen through.

In 2005, the city leased the building to a group led by developer Herbert S. Miller, but a year later, it opted to retain the homeless shelter. Fenty then attempted to develop it but received only two proposals, neither of which panned out.

CoStar relocated its headquarters to L Street from Bethesda after Fenty and the D.C. Council passed a $6 million tax break for the company should it meet certain hiring requirements. Florance said that when he relocated the company it had only about 350 headquarters employees but now has close to 800. It also pays 100 percent of costs for employees who commute using public transit; Florance said almost 80 percent take advantage.

CoStar left some D.C. Council members with a sour taste after Florance flipped the L Street headquarters for a $60 million profit and leased it back, showing off his deep knowledge of the real estate industry.

Council member Jack Evans (D-Ward 2), whose ward includes Franklin, said CoStar was likely to get a fair shake because it had met all the requirements for its L Street building.

“I am looking for something with life,” Evans said. “The last thing we need is for it to remain vacant and closed.”