Ban reported to the Security Council on his controversial decision to temporarily remove the coalition from the UN list of shame, a move that sparked an outcry from human rights groups.
Saudi Arabia reacted angrily to the decision in June to blacklist the coalition after a UN report found the military alliance was responsible for 60 percent of the 785 children's deaths in Yemen last year.
"I still have very strong concerns about the protection of Yemeni children," Ban said, adding that the United Nations is continuing its review with the Saudi-led coalition.
In the letter obtained by AFP, Saudi Ambassador Abdallah al-Mouallimi offered to share with the United Nations the results of 10 investigations of air strikes on hospitals, homes, a wedding party and markets.
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Ban said he had received information on the steps taken by the coalition, but that these fell short and that "the content of the report stands."
"We will continue our engagement to ensure that concrete measures to protect children are implemented," he told a council debate on children and armed conflicts.
"What happened in the past, for me, is behind," said Zerrougui.
Speaking to reporters outside the council chamber, Mouallimi repeated that the de-listing of the coalition was "irreversible, final and unconditional."
"I said that then and it is even more true today."
The coalition has invited UN officials to come to Riyadh to discuss their concerns and to obtain information on the investigations, he added.
In his letter, the Saudi ambassador said the coalition had set up a committee to compensate victims and opened a direct dialogue with aid organizations to guarantee the protection of hospitals.
They include drawing up a list of prohibited targets such as schools and diplomatic missions and working with "local forces to identify and vet targets for airstrikes.