Joseph A. Califano Jr., a veteran Washington lawyer and Secretary of Health, Education and Welfare in the Carter Cabinet, has been asked to serve as special counsel in charge of the House ethics committee investigation into allegations of sexual misconduct and drug use on Capitol Hill.

Califano is considering the offer but has not decided whether to take on the job.

The ethics committee has been empowered to conduct a broad investigation into allegations that some congressmen may have used cocaine or solicited sex with teen-age congressional pages.

Rep. Louis Stokes (D-Ohio), chairman of the panel--the House Committee on Standards of Official Conduct--said Thursday that the committee expected to name a special counsel soon. Stokes could not be reached for comment last night.

The Justice Department is conducting separate criminal investigations into the sex and drug allegations. The criminal inquiries are looking into allegations that some congressmen may have promised pages promotions in return for sexual favors, and that some members may have used cocaine distributed by an alleged drug ring operating on Capitol Hill.

Califano last month completed an 18-month study of drug abuse and alcoholism for the state of New York, and declared that such drugs as heroin, cocaine and alcohol have become "America's No. 1 health problem." Califano supervised the study as a special counselor to New York Gov. Hugh L. Carey.

A partner in the Washington law firm of Califano, Ross and Heineman, Califano has had a long career in public service and politics. He was domestic affairs adviser to President Lyndon B. Johnson, later served as general counsel to the Democratic party and headed up HEW under President Carter until he was fired during a mid-term shakeup of the Carter cabinet in 1979.