Saudi Arabia reacted angrily when the coalition was added to a UN "list of shame" of child rights violators in June, demanding to be taken off.
UN Secretary-General Ban Ki-moon accused the Saudis of threatening to cut off funding to UN aid programs and agreed to remove the coalition from the list pending a review.
That review is due to come up in the Security Council on August 2.
"We hope that in advance of the debate on August 2, the coalition will be able to provide us with some information on the concrete actions they have taken," said UN spokesman Stephane Dujarric.
Saudi Foreign Minister Abel al-Jubeir met with Ban at UN headquarters on Thursday to discuss the situation in Yemen just as peace talks were scheduled to resume in Kuwait on Friday.
During the meeting, Jubeir said the coalition was ready to take "the necessary concrete measures to end and prevent violations against children," Dujarric said, AFP reported.
Jubeir told reporters that the coalition was waging its air campaign "with great care in order to avoid damages to civilians and in particular children."
"We have explained this to the United Nations and our dialogue is ongoing," he added.
A UN official, speaking on background, said the row could be resolved if Riyadh takes action on three fronts: "big movement" in the Yemen peace talks, procedures to lessen civilian casualties and remedial measures.
The United Nations had blacklisted the coalition after concluding in a report that it was responsible for 60 percent of the 785 children's deaths in Yemen last year.
The war in Yemen has killed some 6,400 people and exacerbated the humanitarian crisis in the impoverished country, according to the United Nations.