The shroud of secrecy that Hermann F. Eilts wore into Rome to cover his urgent diplomatic mission of rescuing Egypt's Anwar Sadat from Saudi Arabia's wrath began to unravel last Wednesday when Sudan Airways announced that Flight 113 to Cairo was being canceled.

Eilts had been explaining the complexities of peace in the Middle East to Saudi Arabia's Crown Prince Fahd. But he was suddenly just another stranded airline passenger with appointments he could no longer make, and his absence in Cairo blew apart the embassy's cover story. Word soon leaked he had been on an important mission in Europe.

That mission was part of a larger, continuing U.S. campaign to get the Saudis and Egyptians to stop the bitter public attacks they have exchanged over the past three weeks as Sadat's isolation in the Arab and Islamic world has deepend far beyond the expectations both of Cairo and Washington.

White House officials confirmed last night that as part of the U.S. effort President Carter has written to Saudi Arabia's King Khalid to urge the Saudi royal family once again not to oppose Sadat' efforts. These officials refused to confirm or deny a published report that the letter had specifically asked the Saudis to put up $535 million to purchase 50 F5E jets for Sadat as previously agreed by the three countries.

One presidential aide said that letter contained nothing extraordinary and had been agreed to by national security adviser Zbigniew Brzezinski and Secretary of State Cyrus R. Vance before being sent. This aide indicated that the letter stressed common U.S.-Saudi interests in lessening pollarization in the Arab world and contained "no begging."

But U.S. officials now concede that they have been surprised by the intensity of Saudi opposition to Sadat. Some of these officials are concerned that the Saudis, who were Sadat's principal political allies and bank-rollers until he signed the Camp David peace treaty, are prepared to let the Egyptian leader sink if he runs into serious trouble at home.

These officials do not think the Saudis are actively seeking Sadat's fall. But the royal family is reportedly infuriated by Sadat's slashing attacks and have responded by cutting all new economic aid and suggesting that Saudi financing for the F5E deal may be canceled.

The United States is continuing with plans to sell the 50 jet fighters to Egypt despite the uncertainty surrounding the Saudi financing, State Department officials said yesterday. A Pentagon team is currently in Egypt and will discuss ways to complete the sale as part of the expanding U.S. Egyptian military relationship.

The decision by Saudi Arabia last week to announce with fanfare that it was disbanding the Arab Military Industries Organization, which is based in Egypt and paid for by Saudi Arabia, Qatar and the United Arab Emirates, was one of the sharpest psychological blows to U.S. and Egyptian confidence that the Saudis would eventually renew their support for Sadat and tacitly accept the Camp David peace process.

Egypt has responded by refusing to close down the weapons manufacturing enterprise and is reportedly seeking European financing for new projects.A Radic Cairo broadcast last Thursday called the Saudi economic boycott "illegitimate" and warned the three states that history would judge them harshly "for the attempt to destroy the Arab arms industry."

U.S. officials say they are not certain if the harsh insults Sadat has been aiming at the Saudis since May Day are part of a strategy to rally Egyptian support and show Israel he needs concessions in the negotiations on the West Bank, or if they are imply an impulsive, angry response to Saudi Arabia's lack of support.

The insults and Saudi responses triggered the secret mission last week to Rome by Eilts, one of the department's leading Arabists who is retiring after five years in Cario and who came to know Fahd during a tour in Saudi Arabia.

After reportedly convincing Sadat to tone down the insults, Eilts, who has strode the Egyptian capital as something of a U.S. pro-consul, flew to Italy to meet with Fahd and seek a similar commitment.Informed officials said last night that both sides had in fact muted their polemics in the wake of that meeting.

Embassy officials at first denied to reporters in Cairo that Eilts was out of town but the unexpected delay in returning underminded that denial. Reporters traveling with Vance to London on Sunday were told of the Eilts-Fahd meeting in sketchy terms and could not identify the source. The first account of the Carter letter to King Khalid also emerged from the party of officials and reporters traveling with Vance.

"There was no attempt at anything improper," one upper-echelon State Department official said yesterday. "We were just trying to conduct important business with reasonable discretion." Other officials, chagrined at the disclosures, feared that the publicity could now affect the tentative effort at Saudi-Egyptian rapprochement.