Gov. Bob McDonnell (R) hinted Tuesday that the state may file a lawsuit to get Virginia’s members seated on the the Metropolitan Washington Airports Authority, which is overseeing the construction of the Metrorail extension to Dulles International Airport.

But, McDonnell said, state officials will first try to work out a compromise when all the parties involved sit down with U.S. Transportation Secretary Ray LaHood on Wednesday, May 2.

“We’re trying to work some things out,’’ he said on his monthly call-in show on WTOP.

McDonnell (R) said he’s committed to the project but that MWAA needs to allow Virginia’s members to be seated, keep the costs down and drop a plan to give incentives to contractors who use unionized labor.

“I am a strong supporter of this rail-to-Dulles project,’’ he said.

McDonnell said the law dictates that Virginia’s members be seated, and that the state has minority representation on MWAA, even though the state is paying for the bulk of the project.

“Until we get new accountability into the board, it isn’t representing the people of Virginia,’’ he said.

MWAA officials say they cannot appoint the Virginia members until the D.C. Council passes legislation allowing them to do so.

Last week, the General Assembly failed to add $300 million to the project after the McDonnell administration declined to agree to the additional money. His staff also indicated that the state’s $150 million contribution could be in jeopardy.

“Major movement needs to be made . . . before I could ever envision giving more money,” Transportation Secretary Sean Connaughton told The Post last week. “They’re not going to get the $150 million if they continue to do what they’re doing.”

Connaughton did not return a call for comment Tuesday.

Senate Democrats had wanted bonds to help hold down tolls, which are due to be increased to pay for the second phase of Metro’s new Silver Line.

The first phase of the $6 billion Silver Line is under construction from Falls Church to Reston and is expected to open in late 2013 or early 2014. Construction on the second part of the project, which will run to Dulles and into Loudoun, is expected to start in January and be completed in five years.

To help pay for the project, a one-way trip that now costs $2.25 could increase to $4.50 as early as next year. By 2018, tolls for that one-way trip could rise to $6.75.

McDonnell repeated an oft-mentioned Republican talking point Tuesday, saying former governor Tim Kaine, now a Democratic candidate for U.S. Senate, turned the responsibility over to the authority with no caps on tolls. Kaine did negotiate an agreement on the project at the urging of several Republicans, including then-Sens. John Warner and George Allen and Rep. Frank Wolf.

McDonnell also said the state didn’t have more money to spend on the project right now — and additional funds are not needed until later.

“We want to be able to keep this project on track,’’ he said.