After three years of planning for a new bus terminal in Manhattan, the commissioners of the Port Authority of New York and New Jersey still agree that New York City badly needs one. But, as the nasty debate over paying for it has spilled out in public, they appear to agree on little else.

The discord has threatened to paralyze the board that oversees the agency and that is responsible for transportation projects critical to the region. When the time came on Friday to publish the board’s monthly agenda, the agency punted, hinting that Gov. Andrew M. Cuomo of New York, a Democrat, might instruct his appointees not to conduct any business, just as he did before the board’s previous meeting.

When the agenda for the meeting this week finally appeared on Monday afternoon, it suggested that politicians from the opposite sides of the Hudson River had still not bridged their differences. They may still be billions of dollars apart in their views on how the agency should spend the money it collects from tolls and transportation fees.

The battle over the bus terminal shows just how quickly the Port Authority can fall into the kind of dysfunction that allowed appointees of Gov. Chris Christie of New Jersey, a Republican, to spitefully close lanes at the George Washington Bridge in 2013. Mr. Christie has since stepped back from the agency, but the tussle for influence over its vast finances has remained fierce.