Federal transportation officials floated a plan Thursday to shrink the cost of the Metrorail extension to Loudoun County by building an aboveground station at Dulles International Airport and transferring responsibility for parts of the project to local governments.

U.S. Transportation Secretary Ray LaHood has been mediating discussions between Metro project officials and regional leaders for the past month in an attempt to reach a consensus.

“We’re at a point where we’re making some hard decisions and the aim of the game is to try to make the project affordable for our taxpayers and to our commuters,” said Sharon Bulova, chairman of the Fairfax County Board of Supervisors.

LaHood concluded the meeting Thursday by asking stakeholders to take the federal proposal back to their respective boards for feedback and to reconvene in mid-July.

Participants spoke about the details of the discussion on the condition of anonymity so as not to jeopardize the negotiations.

The most contentious issue for the local, state and federal officials invested in the second phase of the project, now estimated to cost $3.5 billion, has been the location of the airport station. Elected officials, primarily from Northern Virginia, have urged the Metropolitan Washington Airports Authority to back away from its plans for an underground station, which is more expensive but closer to the terminal.

In addition to shifting to an aboveground station, the document presented by Peter M. Rogoff, administrator of the Federal Transit Administration, proposes transferring responsibility for the planned Route 28 rail station to Fairfax County to save an estimated $136 million. Fairfax and Loudoun counties would also take charge of building five commuter parking lots for a savings of $235 million.

One of the main concerns of local officials has been the cost to taxpayers and to drivers using the Dulles Toll Road, who would be on the hook for more than 50 percent of the project’s cost.

The airports authority has maintained that the award of a low-interest loan from the federal government — not the location of the airport station — would have the biggest impact on keeping toll increases under control.

But Transportation Department officials raised the prospect Thursday of providing federal loans to the counties, not to the airports authority.