As a first step, Mr. Cuomo said he and Mr. Trump wanted a precise estimate for the project’s costs, instead of relying on figures from Amtrak, which owns the existing tunnel. State and federal officials planned to discuss how to begin soliciting bids from private companies to examine the costs, Mr. Cuomo said.

Mr. Cuomo had an answer for those critics disappointed that he did not leave the White House with a concrete promise of federal dollars: “If the president wasn’t interested and wasn’t moved by the video, why have the meeting?”

The meeting was attended by Elaine Chao, the federal transportation secretary; Robert Lighthizer, the United States trade representative; John F. Kelly, the president’s chief of staff; and Rick Cotton, the executive director of the Port Authority of New York and New Jersey.

Before President Barack Obama left office, his administration had reached a funding agreement with Mr. Cuomo and Chris Christie, who was then the Republican governor of New Jersey. The two states agreed to cover half of the cost with the federal government providing the other half.

Last year, Mr. Cuomo and Mr. Christie discussed the project with Mr. Trump and Ms. Chao at the White House in a conversation both sides described as “productive.” But Ms. Chao later denied the existence of a funding agreement, leaving the project in limbo.

Mr. Cuomo’s meeting with Mr. Trump came as Congress begins to consider a spending bill that could include funding for the tunnel. On Wednesday, Senator Chuck Schumer, the Democratic leader from New York and a key supporter of the Gateway project, again called on Mr. Trump to help pay for the project, saying failing to do so would lead to “Armageddon for commuters and derail the economy.”

A spokesman for Mr. Schumer said the senator believes Mr. Trump “wants to use Gateway as leverage to trade for the wall.” Mr. Trump campaigned on a promise to build a wall along the souther border of the United States to deter illegal migration.