On Monday, Mr. Sharpton defended her against those who have said she is not qualified for the post. They also spoke by phone, and he proposed a meal at the same table where he met with Senator Barack Obama in November 2007. And on Thursday, after a short session over plates of chicken, collard greens, string beans and lemonade, Mr. Sharpton pronounced Ms. Kennedy, who is pursuing an appointment to the United States Senate, “the unique candidate in this, the unique person in this with some unique skills.”

Ms. Kennedy repaid the compliment, calling their tête-à-tête “fantastic.”

The MTA’s capital construction page shows an update for November with pictures from below, where the agency is hollowing out the cavern for the station and making way for the eventual launch of a tunnel-boring machine, which will slowly dig its way along the 1.5-mile route.

The project, budgeted at $2.1 billion, would extend the line from Times Square to the base of the Javits Center on 34th Street, adjacent to the West Side rail yards. The Bloomberg administration has been the driving force behind the extension, which it says will help spawn tens of millions of square feet of West Side development.

The cash-strapped MTA had no desire to pay for the project, so the city is footing the entire bill, up to the $2.1 billion. Should costs exceed the budget (which many onlookers assume they will, given rising costs everywhere), the city and the MTA have yet to negotiate an agreement on who would cover them.

In dozens of interviews, former aides to the mayor, elected officials, good-government advocates and voters said they have become deeply disillusioned by the way Mr. Bloomberg is corralling support to rewrite the city’s term limits law, which New Yorkers have endorsed twice in citywide referendums.

Over the last three weeks, the mayor and his aides have silenced a potential critic of his third-term bid with the promise of a plum position on a government committee, pressed groups that rely on his donations to speak on his behalf and cajoled union leaders to appear on camera endorsing his agenda.

Those tactics are expected to deliver a victory on Thursday when the City Council votes on whether to allow Mr. Bloomberg to seek a third term. But many of those interviewed say the horse-trading and arm-twisting he has used in pursuing that term are at odds with his claim to being above the fray of rough-and-tumble politics.

. . .

The disenchantment with Mr. Bloomberg runs especially deep among his former aides and advisers at City Hall. In interviews, five of them said they had been surprised and unsettled by the mayor’s tactics. “It stinks of clubhouse politics,” said one former aide. “It’s not like him.”

Another said that when former Bloomberg staff members meet for drinks these days, and the topic turns to his third-term bid, “people roll their eyes and say they are glad to not be there anymore.”

The aides said that they had adopted Mr. Bloomberg’s vision and enlisted in his administration because they believed he was a transformational figure in New York politics.

Richard D. Emery, a civil rights and election lawyer who in the late 1980s helped dismantle the city’s Board of Estimate, which controlled much of the city’s spending, has strongly supported Mr. Bloomberg, describing him as a “terrific politician because he is not a politician.”

“Up until now, he has been a paradigm of what a municipal mayor should be,” Mr. Emery said, but watching Mr. Bloomberg’s heavy-handed approach to remaining in office has left him disaffected, he added.