The Fairfax County Board of Supervisors on Tuesday declined to endorse legislation before Congress that would boost Virginia’s representation on the Metropolitan Washington Airports Authority’s board, with several supervisors expressing fears that the measure would only aggravate regional tensions.

But the Board also asked its legislative staff to draft a letter that would thank its sponsor, Rep. Frank R. Wolf (R-Va.), for his efforts and perhaps pursue other ways for Congress to address the concerns of political leaders in Northern Virginia’s about oversight of the airports authority.

The congressional measure, inserted by Wolf in September in the House’s $55 billion 2012 transportation bill, would increase the size of the airports authority board from 13 to 17 members. Virginia, whose governor currently appoints five members, would gain four more. The others are appointed by the governor of Maryland, the District’s mayor, and the federal government. Wolf’s legislation would also make it easier for political officials to replace their respective appointees.

Supervisor Patrick S. Herrity (R-Springfield), noting that the authority’s two airports are both located in Virginia, urged the Board of Supervisors to back Wolf’s legislation. Saying the measure was a “no-brainer” for increasing the state’s influence on MWAA’s board, Herrity said Virginia taxpayers also deserve more oversight because they are bearing the majority of the costs for building the Metrorail line extension to Dulles.

“It’s our taxpayers. It’s our resources,” Herrity said. “Right now you’ve got a Board that’s serving the interests of Maryland and D.C., and it’s not serving the interests of Virginia.”

Wolf’s legislation comes after a spat between MWAA and local jurisdictions over whether to build a costly underground Metro station, as the airports authority initially intended, or a less expensive, above-ground station, as Fairfax County demanded because of rising project costs. The airports authority eventually reversed course.

But the Board split along partisan lines, with several Democrats expressing reservations about Wolf’s approach.

Board Chairwoman Sharon S. Bulova (D) reminded Herrity that it was Virginia appointees who pushed for the underground station despite local opposition to its cost. Supervisor Linda Q. Smyth (D-Providence) said Wolf’s measure failed to address the belief among local officials in Northern Virginia that they, and not the governor, should have more say over the airports authority.

“Rather than focusing our attention on a hostile takeover, I think we ought to focus our attention on trying to figure out what is the root of the problem, and how do we work it out,” Supervisor Jeffrey McKay (D-Lee) said. “When you see a bill like this introduced, all it does is make our regional partners dig in further. It actually exacerbates the problem we’re trying to fix.”

The board voted 6-3 against Herrity’s proposal, with Supervisor Gerald W. Hyland (D-Mount Vernon) absent. But the board also directed county staff to draft a letter that the supervisors could consider at their Nov. 1 meeting that would seek help from Wolf and other members of Congress on increasing local control over the airports authority.