Working under threat of veto, the House yesterday opened debate on the fate of the Legal Services Corporation, a major program of the nation's War on Poverty which President Reagan has pledged to dismantle.

In a letter circulated to Republican congressmen, White House counselor Edwin Meese III declared that "the administration does not want a legal services bill to be reauthorized" and added he would urge the President to veto the $260 million funding bill.

While a vigorous campaign has been mounted against the program by the conservative caucus and other groups, the powerful American Bar Association, representing the nation's legal establishment, has championed the bill. A petition to Congress this week from 600 state and federal judges in California, including five state Supreme Court justices, urged that funding be continued.

The bill was considerably modified in the Judiciary Committee to placate conservative opponents. It would cut funding by 20 percent as well as prohibit class actions against government entities and legal assistance in abortion cases except where the mother's life is in danger. It would ban litigation to legalize homosexuality or to represent illegal aliens.

The administration wants the federally funded corporation abolished and responsibility for legal aid to the poor transferred to the states. States would have the option of funding legal services from a general block grant for social programs, or they could abandon the program entirely. Private bar associations should provide legal counsel for the poor, the administration contends.

Legal aid proponents, however, charge that with the drastically reduced budget for social programs that is likely to be enacted this year, few states would find the means to provide lawyers for the poor. "It will be a travesty if we vote to deny legal access for the poor, when on the wall of every courthouse in America there is a saying, 'Equal Justice For All,'" said Judiciary Chairman Peter W. Rodino Jr. (D-N.J.)

The bill would fund 323 legal services offices in 23 states, providing one lawyer for every 5,000 poor Americans. By contrast there is one lawyer for every 440 Americans above the poverty line. Legal services attorneys handled 1.5 million matters last year, most of them such routine matters as wills, divorces, tenant disputes, custody fights and other civil cases.

In the House debate yesterday, a Virginia Republican gave one of the more eloquent defenses of the program. The bill, Rep. M. Caldwell Butler said, controls the activities of legal aid attorneys more than ever before and gives Reagan's 11 appointees to the legal services board complete power.

"If block grant money were left to provide legal services, I doubt whether in my state it would be used for that purpose," Butler said. "There is too much competition for that money."

It was perhaps a measure of administration pressure that two Republicans who supported the bill in committee -- Reps. Daniel E. Lungren (Calif.) and Thomas N. Kindness (Ohio) -- opposed it yesterday. Lungren said a woman in his district tried twice to have a legal-aid attorney draft her will. The woman was told there was no time, because it was not a class action, he said.

Rep. Sam B. Hall Jr. (D-Tex.) said legal services brought suit to compel payment of Supplemental Security Income benefits to alcoholics; sued to require the New York City Transit Authority to hire former heroin addicts and, in Hartford, tried to make the state of Connecticut pay for a welfare recipient's sex-change operation.