The Democratic-controlled House bowed yesterday to White House veto threats and agreed to continue special aid for the long-term unemployed only for those currently on the rolls.

Because the benefits last no more than 14 weeks, this would mean an end to the program by early July.

An earlier proposal, dropped in the face of veto threats, would have allowed jobless workers to qualify for benefits for an additional three months. This would have given Congress time to consider program revisions without an interruption in benefits.

The House voice vote approving phase-out legislation came as the Senate Finance Committee approved a similar bill for action by the Republican-controlled Senate today, indicating a strong likelihood of final action on the measure before Congress recesses later this week.

White House officials said President Reagan remained opposed to any continuation of the program, and Senate Finance Committee Chairman Bob Packwood (R-Ore.) told his colleagues that he could not guarantee that Reagan would sign even the phase-out measure.

But officials said Reagan was not necessarily threatening to veto the phase-out bill. They indicated instead that his continued opposition was intended to keep pressure on Congress not to expand the program. Under the program, which began during the recession three years ago, an estimated 325,000 jobless workers receive eight to 14 weeks of extra benefits after exhausting their first 21 weeks of unemployment assistance.

The program has been extended several times since 1982; the last extension expired Monday, with final checks going out this week.

The phase-out measure means that those now on the rolls would receive the full amount for which they are eligible, while those who might qualify next week would be ineligible for benefits.

According to a revised estimate, the phase-out would cost about $180 million, and a full three-month extension would cost up to $440 million. Packwood told the Finance Committee that the administration had earlier underestimated the costs, adding that the higher estimates increase the chances of a veto if the phase-out is broadened.

In clearing the way for floor action on the phase-out, the House Ways and Means Committee dropped without a vote a subcommittee proposal for a three-month extension. A 30-day extension also failed, 19 to 17. The phase-out measure was then approved, 20 to 16.

"It's a question of whether or not we can get a bill signed," committee Chairman Dan Rostenkowski (D-Ill.) said in opposing anything more than a phase-out. A House Democratic aide indicated that Democrats also were fearful of looking like big spenders, saying: "The Democrats are terribly skittish, as they should be, about spending programs."

But Rep. Carroll A. Campbell Jr. (S.C.), ranking Republican on the unemployment compensation subcommittee and sponsor of the phase-out, appeared adamant about continuation of benefits for those on the rolls.

"We're prepared to stand against the administration if they don't support this," he said.

In the Senate Finance Committee, a proposal to extend the program for three months failed, 11 to 2, with Sens. John Heinz (R-Pa.) and Daniel Patrick Moynihan (D-N.Y.) supporting it. The phase-out then was approved unanimously.

"It's a question of entire communities going out of business" if supplemental benefits are not continued for areas with high unemployment, Heinz argued.