The Legal Services Corp., under pressure from Congress, has agreed to drop a proposed rule that would have made it much harder for poor clients to qualify for legal aid.

But the corporation has refused to abandon another controversial proposal that would make it virtually impossible for advocacy groups to hire legal service attorneys.

Rep. Mario Biaggi (D-N.Y.), chairman of the House Aging Committee's subcommittee on human resources, said corporation Chairman Donald P. Bogard agreed in a letter yesterday to drop the tougher eligibility criteria for poor clients. The corporation had announced the proposed "national means test" on Aug. 29.

Under the criteria, the 336 local groups that receive grants from the corporation would have had to turn away clients who had more than $15,000 equity in a house or $4,500 in an automobile, unless they had substantial debts or medical expenses.

The regulation also would have cut off services to anyone whose income was more than 187.5 percent of the national poverty level; that level is $9,862 for a family of four. The proposed rule contained a number of exceptions allowing local offices to deny services to clients with incomes lower than that.

Biaggi claimed the criteria would have eliminated "one-half to two-thirds of elderly persons currently eligible for legal services." He said Bogard agreed to abandon the criteria after more than a dozen groups that represent elderly and disabled persons criticized the plan at a heated subcommittee hearing last week.

Biaggi said the corporation has decided to continue to allow local legal service offices to establish their own eligibility guidelines.

The corporation, however, has refused to withdraw proposed rules that would impose an assets test on members of advocacy groups.

Currently, the corporation considers the assets of the group receiving legal representation when it determines eligibility, Biaggi said. For instance, if an advocacy group represents nursing home patients, the corporation now looks only at the patients' assets. Under the proposal, the corporation also would consider the assets of the advocacy group representing the patients.

Biaggi contends that the rule will "greatly inhibit group representation on behalf of older people such as those in nursing homes." Sen. John Heinz (R-Pa.), chairman of the Senate Special Committee on Aging, joined Biaggi yesterday in demanding that the corporation drop both proposals. Bogard told Biaggi the revisions will be presented to the full Legal Services board on Oct. 4 for action.

Last month, John C. Meyer, the corporation's deputy general counsel, said the rule changes had been proposed because of the agency's shrinking budget. "It will make a difference in concentrating our resources on the poorer people," he said. "But this is not really major."

The Reagan administration took office hoping to abolish the corporation, which became a thorn in the side of many local and state governments in the process of developing a reputation as a feisty advocate of the poor. Its current budget is $241 million, down from $321 million in fiscal 1981. It has requested $257 million for fiscal 1984.