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Gov. Chris Christie of New Jersey met with the federal transportation secretary, Anthony Foxx, on Aug. 18 to discuss plans to build a new tunnel, known as the Gateway Project.Credit Mel Evans/Associated Press

The governors of New York and New Jersey sent a letter to President Obama on Tuesday offering to line up funding for half of the costs of new Hudson River rail tunnels if the federal government pledges grants for the other half.

In the letter, Govs. Andrew M. Cuomo of New York and Chris Christie of New Jersey emphasized the importance of building the tunnels and committed to paying for their states’ “fair share” of the project, which could cost as much as $20 billion.

“We are writing jointly in an attempt to move the stalled project forward by putting a funding proposal on the table that we believe is realistic, appropriate and fair: split the responsibility for the cost,” they said in the letter.

On Tuesday night, the White House issued a statement saying the federal transportation secretary, Anthony Foxx, would work with the two states to determine how the federal government could “best support progress on this project.”

“We are encouraged that all parties have come to the table committed to move the project forward,” Frank Benenati, a spokesman for the White House, said in the statement.

The letter from the governors did not say how the states would raise their share of the project’s costs. The governors suggested that the regional effort be led by the Port Authority of New York and New Jersey, an agency that has been under legal and political scrutiny for more than a year amid criminal and legislative investigations. Last week, the chief executive of United Airlines resigned as federal officials examined whether the airline traded favors with the authority’s then-chairman, David Samson.

Still, state officials and business and transportation groups praised the letter as a breakthrough in efforts to advance the tunnels. It came a month after Mr. Christie, a Republican who is running for president, met with Mr. Foxx and agreed to work with him on a plan to build the tunnels, known as the Gateway project.

A crush of rail delays in July brought renewed attention to the century-old tunnels that run under the river and need repairs because of damage caused by Hurricane Sandy in 2012. Mr. Foxx has said the Gateway project was one of the most important rail projects in the country and called the lack of action to repair the tunnels “almost criminal.”

He said on Tuesday that he would work quickly with local officials on the details of paying for the project, with the goal of “an equitable split between the states and the federal government using our existing grant and financing programs and whatever new resources can be provided by Congress.”

Both governors have recently expressed support for building the tunnels. But the letter was the first time they had voiced a specific commitment to fund the project. And it signaled a more conciliatory tone after the sometimes acrimonious rhetoric of the last few months. In August, Mr. Cuomo, a Democrat, told reporters, “It’s not my tunnel,” arguing that the current one was owned by Amtrak and used by Amtrak and New Jersey Transit.

On Tuesday, Mr. Cuomo said that he hoped the offer would jump-start work on the project. “We’re hoping to break the gridlock by making what I think is more than a good-faith effort to say 50-50,” he said. “Otherwise I’m afraid this is going to go on forever.”

In New Jersey, where many residents rely on the tunnel to reach their jobs in New York City, Mr. Christie has received criticism for his decision in 2010 to cancel a plan to build tunnels under the river, known as Access to the Region’s Core. At the time, he argued the plan placed an unfair financial burden on taxpayers in his state, but has said in recent months that he would support building the tunnels if the federal government provided appropriate funding.

Senator Chuck Schumer, Democrat of New York, has proposed creating a nonprofit development corporation to oversee the project and bring together different sources of funding. The governors proposed creating a dedicated staff and an entity at the Port Authority to plan the project.

Veronica Vanterpool, executive director of the Tri-State Transportation Campaign, an advocacy group, said the governors had moved beyond “empty rhetoric toward earnest commitment” and called for reforms at the authority. “This project is too important to get mired in scandal,” she said in a statement.

Amtrak officials and local leaders applauded the governors’ letter. Kathryn S. Wylde, president of the Partnership for New York City, a business group, called the project essential to the region’s economic future.