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Andy Byford, the transit executive who was hired to rescue New York City’s floundering subway, and Gov. Andrew M. Cuomo have increasingly clashed over management of the system, and several of Mr. Byford’s colleagues said they feared he might quit.

The two men did not speak between January and April, even as Mr. Byford was seeking to move forward on a sweeping $40 billion plan to overhaul the subway in the next decade.

If Mr. Byford, who was hired in November 2017, were to step down, it would be a major blow to efforts to improve the system, which has been plagued by antiquated equipment, cost overruns and rising complaints from riders about chronic mismanagement. In recent years, New York’s subways have had one of the worst on-time rates of any major rapid transit system in the world.

Mr. Byford and Mr. Cuomo have disagreed over the plan to fix the L train, new technology to upgrade subway signals, the high cost of Mr. Byford’s “Fast Forward” overhaul plan and Mr. Cuomo’s regular criticism of the authority.