For the third time in three months, the District of Columbia and the U.S. department of Labor and Health, Education and Welfare exhausted their legal power to spend money yesterday, and Congress decided to wait until next week to seek a solution.

Unless the Labor Department's spending authority is restored by Wednesday afternoon, a spokesman said, that agency's 17,000 employees - about half of them in metropolitan Washington - will receive pre-Christmas paychecks short by 20 per cent. HEW has until next Friday morning and D.C. until mid-December before facing pay problems, officials said.

House Majority leader Jim Wright (D-Tex.) announced yesterday that the House will consider the issue Tuesday. A solution would involve either a single "continuing resolution" to provide stopgap funding for all three governmental units, or two resolutions - one for D.C., another for Labor and HEW.

An extension of funding for Labor and HEW, once it is approved by the House, faces a filibuster threatened in the Senate over the emotional issue of abortions paid for by HEW's Medicare program.

At present, the only measure pending in the House would extend the D.C. spending authority until Sept. 30, 1978, at the same rate as in the last fiscal year. If Labor and HEW are attached to the D.C. bill, the city's finances would hinge upon a solution of the abortion dispute.

Funding for all three governmental units has been on a month-to-month basis since Oct. 1, the start of the fiscal year. The most recent continuing resolution expired Wednesday night.