The United States "deeply regrets" what it considered the "necessary" interception of an Egyptian aircraft with the hijackers of an Italian cruise ship aboard, according to a statement by U.S. Ambassador Nicholas Veliotes here today.

Reading prepared remarks to a group of reporters summoned to the embassy this afternoon, Veliotes also praised Egypt's actions in the ending of the hijacking of the Achille Lauro cruise ship Wednesday.

"We have no interest in offending the government or people of Egypt," Veliotes said. "We are friends."

Egypt is America's closest ally in the Arab world and the only Arab country formally at peace with Israel. The United States supplies Egypt with billions of dollars in military and economic aid each year.

But yesterday, as Cairo saw violent clashes between angry students and police, President Hosni Mubarak said in a bitter talk to reporters that he was "wounded" by the U.S. action and "to be able to overlook this, it will take a long time."

Veliotes spoke after delivering a letter from President Reagan to Mubarak that U.S. officials described as "a very good first step in the aftermath of this" to move toward smoother relations as quickly as possible.

But even as the United States sought to heal the wounded relationship, Egyptian officials expressed worries about broader regional implications. Mamdouh Beltagy, a senior government spokesman, suggested that what he called the American "hijacking" of the plane and the impending trial of the terrorists in Italy will only make the cycle of violence in the Middle East more vicious.

Beltagy said in an interview that the original agreement ending the hijacking had assured the terrorists of safe passage to the country of their choice.

But when Egyptian officials learned subsequently that the Palestinian terrorists had killed an elderly American invalid, Beltagy said, they decided to send them to Palestine Liberation Organization Chairman Yasser Arafat for prosecution as "a way to test Arafat's political will against terrorism."

The current peace process, in which Egypt is deeply involved, hinges in part on Arafat as leader of the most moderate elements of the PLO, which is formally recognized by Arabs as the sole legitimate representative of the Palestinians.

While the PLO continues to carry out military actions against Israel, its officials insist that its fighters act only against military targets.

The hijackers were handed over to the PLO because Arafat said he would indeed try them, and at the time of the decision, Reagan had endorsed this idea, Beltagy noted. The White House quickly modified Reagan's statement.

"We believe that if they the terrorists are judged by Italians or Americans, it will open the way to another round of retaliation and a further escalation," Beltagy said.

"This is a better way to cut short this vicious circle of violence, judgment, retaliation and so on," said Beltagy.

Western diplomats here said today that for similar reasons they believe Egypt did not want the responsibility of trying the hijackers and risking reprisals or heightened domestic violence.

"We'll never know if Arafat would have carried out his claim," said a U.S. diplomat, who asked not to be quoted by name. But "clearly we didn't think that the chances of his carrying out his claim to try them were very good."

It is still not clear whether the hijackers were directly connected with the PLO, but they identified themselves as members of one of its factions, the Palestine Liberation Front, and appeared to confide in that group's leader, Mohammed Abbas.

Abbas, who played the central role in negotiations to end the hijacking, has been accused by the United States of masterminding it. Washington issued a warrant for his arrest after he landed in Italy and continued its quest to obtain control of him in Yugoslavia. Belgrade has indicated that, like Italy, it is not inclined to cooperate.

Abbas was interviewed today in Belgrade by the official Egyptian news agency. The PLF leader confirmed reports by other Palestinian officials and, earlier, by Israeli intelligence that the hijack was the result of an aborted mission to land the four Palestinians in the Israeli port of Ashdod, the ship's next scheduled stop after Egypt.

"An accident changed the course of events when their aim was revealed," Abbas said.

According to reports from an Italian news service, a member of the crew stumbled onto one or more or the Palestinians cleaning their guns. Other crew members and passengers said after the hijacking that they had been suspicious of the four.

Abbas did not concede that he in any way directed the operation, and American officials here presented no direct evidence that he had played such a role.

The Egyptian plane returned to Cairo from Italy today and the ordeal of the ship ended when the Egyptians announced that their investigations were complete and allowed it to sail for Naples from Port Said.