The Metro board, responding to a threatened cutoff of federal spending, appealed to President Reagan yesterday to restore funds in his next budget to help complete "America's Subway."

Gladys W. Mack, outgoing chairman of the transit authority's policy-setting board, warned that the proposed spending halt would result in "serious economic dislocations" because of its impact on the Green Line, the rail system's only unopened route. She asked Reagan to give "personal attention" to the issue.

Last month, the Office of Management and Budget recommended cutting off all funds for Metro construction after the current fiscal year as part of an effort to reduce federal deficits to comply with landmark balanced budget legislation, known as the Gramm-Rudman-Hollings measure.

Federal mass transit officials initially sought to overturn the budget agency's recommendation, according to an internal memo made available to The Washington Post. The Dec. 13 memo urged a "conditional appeal." However, sources said, the move was dropped and no appeal was submitted to OMB.

Ralph L. Stanley, the Reagan administration's mass transit chief, later said the White House was virtually certain to uphold the budget agency's recommendation. The proposed cutoff has threatened to disrupt Metro's plans to expand the rail system to 89.5 miles by the early 1990s.

"This proposed withdrawal of funding could not come at a worse time, because of its impact on the Green Line," Mack, an aide to D.C. Mayor Marion Barry, said in a letter to Reagan. "This would be the first time any administration has 'zeroed out' Metrorail," she added.

"We've faced crises over the years," warned Cleatus E. Barnett, who replaced Mack yesterday as Metro board chairman. "This may well be the most serious."

White House spokesman Dale Petroskey declined to comment on Mack's letter, saying that the White House had not received it. The White House also "traditionaly makes it a policy of not commenting on budget matters prior to the release of the budget," Petroskey added.

Stanley has said it appears likely the Metro system can be expanded to about 76 miles with funds previously appropriated by Congress. An earlier 76.4-mile plan included a Green Line section between Anacostia and a U Street station in Northwest Washington.

Metro officials now are weighing the issue.

In other developments:

*The Metro board approved a controversial multimillion-dollar plan to provide special lifts on half the transit system's buses by the mid-1990s to accommodate disabled passengers.

*The board adopted a series of measures aimed at reducing friction between the Metrobus system and local governments that operate their own bus systems. The moves include changes in a long-debated formula governing subsidies paid by localities for Metrobus service.

*Forrest T. Gay III, 52, a retired Army Corps of Engineers brigadier general, was named to the $71,500-a-year Metro post of assistant general manager for design, construction and facilities maintenance.