The tensions, which are also connected to Saudi-Qatari competition for influence among Syrian rebel groups, escalated following last year's overthrow of Egyptian President Mohammed Morsi's Brotherhood-backed government.
Qatar gave billions of dollars in aid to Morsi's government before it was overthrown by the military amid massive protests against his yearlong rule.
Saudi Arabia and other Gulf states have since stepped in with billions of dollars in aid for the military-backed government.
Details of the Gulf agreement, which also involved Kuwait, the United Arab Emirates and Bahrain, have not been made public.
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"I appeal to the people and leadership of Egypt to seek with us the success of this step in the march of Arab solidarity," King Abdullah said in his first public comments since the Sunday accord. His remarks were carried by the Saudi Press Agency.
Egyptian President Abdel-Fattah el-Sissi's office immediately responded, saying that Egypt would spare no effort "to support its brothers." It called on regional opinion leaders and journalists to help "heal the rift." Saudi Arabia, the UAE and Bahrain had withdrawn their ambassadors from Qatar in March after Egypt recalled its envoy.
Following several rounds of high-level talks and months of pressure, Qatar recently expelled top Brotherhood figures who had been based there. Egypt had wanted some handed over for inciting violence.