Prospects that President Carter may delay his 11-country foreign tour next month were dramatically increased yesterday with the suggestion by Senate Majority Leader Robert C. Byrd that the President will be needed in Washington to lobby for his energy program.

Byrd said he has grave doubts that the Senate and House will be able to complete action on the energy package before Nov. 22, the day Carter plans to leave on one of the most ambitious trips every taken by a U.S. President.

The West Virginia Democrat stopped short of calling for Carter to postpone the trip. But, he told his weekly news conference, "psychologically, it would be a good thing."

Carter has previously indicated he would delay the trip if his energy program were in trouble. And the White House yesterday reiterated that position, stating the administration "remains hopeful we won't have to do so."

The Senate is moving toward passage of the last portion of Carter's five-part energy package. This, Byrd said, should occur early next week. A lengthy controversy then is expected between House and Senated conferees as they attempt to resolve major differences between each house's version.

"When it gets in conference, I think the administration has a very important role to play," Byrd said. Carter's presence in Washington during this period "would indicate he puts a great priority" on the energy program, he added later.

In other matters, Byrd warned during the news conference that U.S. military aid to South Korea may be cut off if the Seoul government fails to cooperate in the investigation of influence-buying in Congress.

"I don't think the Tongsun Park matter should influence our votes on the President's proposal to transfer arms to South Korea," he said. "But practically speaking I think it will. We have already seen that."

The House has scheduled a vote Monday on a resolution, endorsed unanimously last week by its International Relations Committee, demanding cooperation from the South Korean government in the investigation.It is expected to pass.

The House last month came within 15 votes of cutting off aid to South Korea because of its refusal to cooperate in the probe. And International Relations Committee Chairman Rep. Clement Zablocki (D-Wis.) has said it is probably impossible for the Carter administration plan of transferring $800 million worth of military equipment to South Korea to win congressional approval unless Seoul forces Park to testify.

Byrd said this would be a serious mistake, which, would endanger vital U.S. interests all over Asia. "The stability of South Korea is important to that whole area and important to ourselves and that stability can be best enhanced by continuing cooperation and mutual understanding between the United States government and the South Korean government."

"You cut off military aid to South Korea and we still don't get Tongsun Park, what have we done?" he added. "I don't think we should let one situation get in the way of other."

Carter's foreign trip is scheduled to include stops in eight countries on four continents, including the first state visit by an American President to a black African nation.