A system of congestion pricing in Manhattan could be up and running in two years if New York state lawmakers approve it as part of this year’s budget, top officials of the Metropolitan Transportation Authority said Tuesday.

MTA President Patrick Foye said he was confident his agency could create a tolling system in less time than other cities. He and other MTA leaders support New York Gov. Andrew Cuomo’s proposal to charge vehicles that enter Manhattan south of 60th Street. Previous congestion-pricing plans have applied to...