A battle is brewing between New York and New Jersey over who will pay for the billions of dollars in expected cost overruns at ground zero.

Governor Corzine through a spokesman said yesterday he was “dismayed” a day after the Port Authority of New York and New Jersey announced it could not say when construction at the World Trade Center site would be completed or how much it would cost. The spokesman for Mr. Corzine, Jim Gardner, said the governor “does not want to see these cost overruns and delays become an excuse to reduce funding commitments to projects that are already in the capital plan.”

According to a number of sources, Mr. Corzine is sending a clear signal to both the Port Authority and to Governor Paterson: Keep your hands off the funding that is already in place for the more New Jersey-friendly transit projects — specifically the ARC Tunnel project and the PATH transit hub.

The Port Authority, which owns the 16-acre World Trade Center site, is a bistate agency and Messrs. Paterson and Corzine hold equal sway over its leadership. The governor of each state appoints six members to the board of commissioners and both governors retain the right to veto the actions of commissioners from their respective states.

“To hear Governor Corzine speak out proves that New Jersey, which is in dire need of money and doesn’t have hundreds of quasi-government agencies like New York does, wants to make sure they get their share of the swag,” a former executive director of the Port Authority, George Marlin, said in an interview. “They must be getting nervous that this money they are depending on may be in jeopardy.”

On Monday, the executive director of the Port Authority, Christopher Ward, said an accurate cost and scheduling assessment at the site was impossible due to at least 15 unresolved decisions.

Mr. Ward said he would be providing an assessment that includes dates and possible budget overruns by the end of September. The current projected budget, which includes five new office towers, a memorial museum, a park, a performing arts center, and a transit hub, is about $15 billion; the total price tag could increase by $3 billion or more, according to published reports.

A Port Authority source said Mr. Corzine is positioning himself in anticipation of the September report.

“Corzine doesn’t want to be seen as being willing to take away from projects that are closer to New Jersey, like the ARC tunnel and the PATH train station,” the source said.

The $7.5 billion ARC Tunnel project — referring to “access to the region’s core” — would create a new passenger rail tunnel under the Hudson River between New Jersey and Manhattan. Current funding plans include $3 billion from the Port Authority, with the remainder coming from New Jersey and the federal government.

Yesterday, Mr. Ward announced that the Santiago Calatrava-designed PATH transportation hub would not maintain the function that would allow the structure’s wings to open and close. Mr. Ward said the decision allows for the station to be built sooner and for the Port Authority to save money.

“This is the kind of tough decision that we will face,” he said to reporters at a breakfast in Lower Manhattan.

When asked who would pay for the ground zero cost overruns, Mr. Ward, who was appointed by Mr. Paterson, said he expected all the existing partners in the project to chip in.

“This is a project that has many partners,” he said, pointing to the Federal Transit Authority, the federal government, the Port Authority, Larry Silverstein and private stakeholders. “When there is going to be budget issues going forward it will be the same team that answers that budget question that answers the engineering questions.”