With the clock about to run out on federal highway funding, House Speaker John A. Boehner and Senate Majority Leader Harry M. Reid sent a message Tuesday to their members: Find a way to compromise.

“Senate leader Harry Reid and Speaker John Boehner have told Chairman John Mica and me to finish our work this week on the transportation bill,” Sen. Barbara Boxer (D-Calif.) said after a Capitol Hill meeting of the four leaders. “I have asked Chairman Mica to meet continually over the next several days to achieve this deadline.”

Without a conference-committee deal by June 30, Congress will face extending funding levels set in 2005 yet again. An extension may force Congress to deal with an issue it has avoided for almost three years: how to pay for the nation’s long-term transportation needs.

There have been nine extensions since the last long-term transportation bill expired in 2009, and divisions this year have been particularly pronounced in the House. After the Senate approved a bill with bipartisan support, a House bill failed amid Republican infighting over transit funding.

The root problem is bigger than partisan differences. Even when Democrats held both houses, Congress couldn’t decide how to deal with funding. At present, the federal government provides about $40 billion of the $160 billion spent on transportation each year.

But the funding mechanism that built the interstate system and paid the federal share of transportation money for more than half a century doesn’t work anymore. Unless Congress agrees on the bill now in conference, by early next year Washington won’t be able to pay up when states send in bills for federally approved projects, according to the nonpartisan Congressional Budget Office.

The bill the Senate sent to conference committee doesn’t solve the problem, either. It uses several bandaids drawn from elsewhere to shore up projected shortfalls in the Highway Trust Fund. That money is a one-time fix, not a revenue stream.

That bill, everyone agreed, was “buying time” for Congress to address the real problem.

That problem is that the 18.4-cent federal gas tax isn’t enough to pay the bills, and if Congress continues to spend at current levels it will be between $85 billion and $115 billion in the hole by 2021. Raising taxes of any sort has been seen as political suicide.

The other most talked about option — charging people for the miles they travel, through some sort of toll — is profoundly unpopular among those who argue that Americans already have paid for the interstate system and should not be made to pay again.

Current law requires that if the trust fund runs low, Washington must ration payments to states to keep the fund in balance. States pay up front for federally funded projects and submit bills. If a trust fund shortfall causes Washington to become a slow-pay, each state would have to decide how long it could foot the bill before shutting down highway and transit projects.

“We would project that for a six-to-eight-week period, we could float the cash for projects that already are underway,” said Jack Cahalan, spokesman for Maryland’s secretary of transportation. “However, with any slowdown in reimbursements, we certainly wouldn’t advertise any new projects. We’d put a hold on those.”