Some board members have criticized Mr. Cuomo’s secrecy in announcing the new approach without their input. The board must eventually approve the new plan, though no vote was taken on Tuesday.

A spokeswoman for the governor, Dani Lever, said in a statement: “The same consultant studied both options and said again today at a public hearing that this is a different design than the first option they studied, and hence they have a different opinion, just like an apple is not an orange.”

At the board meeting, Polly Trottenberg, the city’s transportation commissioner, asked how the agency would handle silica dust and how long the construction work would last. Leaders on the project said efforts to contain the silica dust would depend on how much concrete is removed. The work could last 15 to 20 months, though the schedule is still being determined.

Ms. Trottenberg, who is also a board member, criticized the agency’s decision to announce “shutdown averted” before talking to the board.

“Is the decision made?” she asked. “Do we have any actual role here?”

The authority’s chairman, Fernando Ferrer, dared board members to express support for a full shutdown — an option that has outraged L-train riders.

“Look, if you’re for inconveniencing 275,000 people, say so!” Mr. Ferrer said.

The M.T.A. has become a punching bag over the last two weeks. Mr. Cuomo, complaining of its stodgy bureaucracy, said he wanted to “blow up the M.T.A.” A piece on New York Magazine’s website asked: “Is the Metropolitan Transportation Authority staffed by idiots?”