The two biggest airlines at O’Hare Airport say they’re moving forward with a lawsuit to block the City of Chicago from building new runways there.

In a statement, United and American Airlines say they failed to reach an agreement with Mayor Richard Daley during a meeting Wednesday morning in Washington, D.C. The meeting was hosted by U.S. Transportation Secretary Ray LaHood, a former Illinois congressman who is trying to mediate the impasse.

The airlines would be on the hook for some of the multi-billion dollar cost of the expansion project, which would add two new runways and reconfigure others in order to cut down on delays. But they’ve said that current air traffic numbers don’t justify the “staggering” expense. And the CEOs of both companies apparently weren’t convinced otherwise Wednesday morning, despite meeting face-to-face with Daley at a meeting brokered by the nation’s top transportation official.

Mayor Richard Daley said in a statement of his own that all sides had a “candid conversation.”

“Now, members from the United States Department of Transportation, the Federal Aviation Administration, the airlines and the City continue to meet and I look forward to a solution that will benefit everyone involved,” Daley said.

United and American left the door open for the project to continue by other means.

“[We] did agree to meet with city and federal officials to explore the opportunity for federal funding” of the runway portion of the project.  The statement did not mention ways to pay for a costly new Western Terminal, which had been part of the original project.

Representatives for the mayor’s office didn’t immediately respond to requests for comment on the airlines’ statement.

A spokesman for the airlines wouldn’t elaborate on the possibility of using federal dollars for the runway work, and a spokesman for the Department of Transportation declined to offer any details on the meeting.

The Daley administration maintains construction is already years behind schedule, and it wants to break ground this spring.

The airlines’ statement also said they’ve agreed to meet with city and federal officials to “explore the opportunity” of getting federal money to help finish the expansion.

United and American filed a lawsuit against the city last month, saying Chicago tried to make an illegal end-run around the airlines when the city moved to borrow money to pay for the final phase of the expansion project, without first seeking airline approval. The airlines must sign off before the city moves ahead with any capital projects at O’Hare, the companies have argued in court documents.