By 3:15 p.m. on Tuesday, Mr. Cuomo’s team — Joseph J. Lhota, the chairman of the M.T.A., and Pat Foye, the agency’s president — claiming they were caught off-guard by the first call, organized their own conference call and were on the line playing defense.

It is “disingenuous” and “false,” Mr. Lhota said for the city to claim the governor’s plans were unilateral. The city, Mr. Lhota pointed out, has final veto power over the transit portion of the M.T.A.’s capital plan. “They have the ultimate leverage,” he said, “the ability to say no.”

At a news conference on Tuesday in the Bronx, Mr. de Blasio weighed in on the latest skirmish. “We’re going to continue to make the point that we don’t like the direction the M.T.A. is taking, and we’re going to be speaking up about it,” he said. “The countdown clocks and the Wi-Fi and painting, having lights on bridges — all that stuff doesn’t matter compared to your subway actually arriving where it’s supposed to arrive on time.”

The governor’s office declined to comment on the interactions between the two men.

The bad blood between the governor and the mayor has included disputes over prekindergarten financing to a renegade deer apprehended in a Harlem building complex. But their protracted battle over the subway seems to have eclipsed all the others, even as riders complain that they have yet to see the kind of improvements Mr. Cuomo promised after he declared state of emergency on the subway last year.

And the schism over the station overhauls is but a piece of the mayor’s differing vision on the subway. The governor’s proposal to allow special tax districts on developments near the subway as a way to help pay for the system is too broad in scope, according to the mayor’s aides, and was done without consulting the city. During an appearance on a morning talk show on Wednesday, the mayor touted his plan to tax the richest New Yorkers, a so-called “Millionaires Tax,” to fund the subway, as more fair that Mr. Cuomo’s notion to pay for it through congestion pricing, charging drivers a fee to enter Manhattan’s business districts.

On Wednesday, the full M.T.A. board met at the agency’s headquarters in Lower Manhattan, and both camps came prepared. Late the night before, Mr. Lhota sent out an open letter. “The city claims no financial responsibility for the subway system that it owns and polices and is the lifeblood of the city’s economy,” he wrote. “The mayor’s answer is simple — and he should just say it — he doesn’t want to fund the subways and help riders. So be it.”

Ms. Trottenberg came to the board meeting equipped with a handout: a list of subway stations the city said needed improvements compared with the governor’s 33 selected stations — only three overlapped. (Her demand for an explanation about how the governor chose the stations slated for improvement were partially answered by Mr. Lhota, who said the stations were selected in part on the ease with which the improvements could be made to figure out best practices.)