D.C. Del. Walter E. Fauntroy plans to ask Palestine Liberation Organization leader Yasser Arafat in Lebanon today to help stop the violence in theMiddle East and recognize Israel's right to exist.

An aide said last night that Fauntroy, who arrived yesterday in Beirut, also hopes to go to Israel to make a similar request of Israeli Prime Minister Menachem Begin.

Fauntroy, as chairman of the Southern Christian Leadership Conference, is visiting Lebanon with SCLC President Joseph Lowery at the invitation of the PLO. The visit resulted from their meetings in New York last month with PLO and Israeli representatives during the furor over the resignation of United Nations Ambassador Andrew Young.

Some Jewish leaders here have described themselves as "deeply disturbed" over Fauntroy's role in the initial meeting with the PLO representative. It was believed that the current plan may arouse renewed concerns on the part of some Jews, who consider the PLO a terrorist group.

A spokeman for Fauntroy said he knew of only two calls made to his office here yesterday about the trip. One caller said she was proud of what she considered Fauntroy's role in working for peace. The other appeared troubled by plans to meet with the chief of what the caller viewed as an organization espousing terror.

State Department and White House spokesmen said last night that they had no comment on the trip which, according to Fauntroy's aide, is being paid for by the SCLC.

"This is not a grandstand show," Fauntroy's press spokesman, Eldridge Spearman, said last night of the SCLC leaders' Middle East trip. "This is a very serious mission they've undertaken in the name of peace."

He said Fauntroy and Lowery "are serious" about continuing with Begin and Arafat the dialogue that began in New York with their representatives.

Spearman said they plan to take advantage of the meeting with Arafat to ask him to "stop the violence and stop the killing and recognize Israel's right to exist as a nation."

At the same time, he said, they hope to meet with Begin to "say to him please stop the violence and warfare and recognize the human rights of the Palestinians as a step to peace." However, the hoped-for meeting with Begin was reported uncertain last night.

In addition to Arafat, Fauntroy and Lowery plan to meet today with Lebanese President Elias Sarkis, the aide said.

In his arrival statement yesterday, Lowery was quoted by the Associated Press as saying: "This trip should indicate to you our support for all the people in the Middle East and or (the) right to a homeland for all those who do not have a homeland."

The 10-member SCLC group reportedly conferred after their arrival in Beirut yesterday with a senior PLO official. Details of the meeting were not available.