The White House has offered fresh assurances of support to Saudi Arabia's royal family to dispel Saudi fears of a rift with Washington, U.S. and Saudi sources report.

The reassurances were conveyed by White House aides last week to Prince Bandar ibn Sultan, son of Saudi Arabia's defense minister and one of the family's most experienced students of U.S. politics. Bandar came to Washington to meet U.S. officials after the Saudis became upset over what they saw as possible signals of U.S. displeasure with their policies.

Despite appeals from the Carter administration, the Saudis are publicly opposing the Egyptian-Israeli peace treaty. Saudi Arabia joined other Arab countries in breaking off diplomatic relations with Egypt this week.

Among the things triggering Saudi concern, according to U.S. and Saudi sources, were an article in The Washington Post about recent U.S. intelligence reports that question Saudi political stability and a Senate staff study that downgrades Saudi Arabia's oil production capability and thus its long-term strategic value to the United States.

In meetings that included a session with David L. Aaron, President Carter's deputy assistant for national security affairs, Bandar was assured that the administration's support for Saudi Arabia was not wavering and that the White House was not trying to send signals via the press or the Senate, according to the sources.

The fact that the Saudis sought the meetings at all indicated the frayed lines of communication that have developed in recent months between Riyadh and Washington, a Saudi source acknowledged.

Confusion about high-level visits has helped spark some of the problems.The Saudis were upset last January by what they saw as demands for oil production increases relayed by Secretary of Commerce Juanita M. Kreps during a visit. This spring a visit by Secretary of Defense Harold Brown failed to produce agreement on a joint military commission or a regional security role for the United States proposed by Brown.

The most serious incident was the cancellation of a scheduled visit to Washington by Crown Prince Fahd. The Saudis indicated the visit-which would have brought Fahd to Washington about the time of a renewed effort by Carter to get Egypt and Israel to sign the peace treaty-was canceled for political reasons, while the White House insisted that Fahd was ill.

This public assertion deeply distressed the Saudi royal family, which dislikes attention being drawn to the health problems of the heir to the throne of King Khalid, who underwent open-heart surgery in the United States last year.

But a trip by Prince Fahd this month to Psain has been officially described as being for health reasons. This time the Saudis were upset when U.S. intelligence reports speculating that the trip might be for political reasons surfaced in The Post.

Officials from both governments say they now feel that efforts to improve communications are having some effect. The Saudis have named a career diplomat, Abdel Aziz al Hegelan, as their new ambassador here, a move that pleases State Department experts.

The Saudis have also invited a small group of prominent American educators, journalists and business figures to tour Saudi Arabia next week.