UN officials plan to travel to Riyadh to obtain more details on various issues, such as rules of engagement, one of the sources said.
A UN annual report on children and armed conflict said the coalition was responsible for 60 percent of child deaths and injuries in Yemen last year, killing 510 and wounding 667. The Saudi-led coalition includes United Arab Emirates, Bahrain, Kuwait, Qatar, Egypt, Jordan, Morocco, Senegal and Sudan.
UN Secretary-General Ban Ki-moon temporarily removed the coalition from the blacklist - contained in an annex to the report - on June 6 pending a joint review after Saudi Arabia, a key UN donor, threatened to cut funding to the world body. Riyadh denied using threats.
Ban is to brief the UN Security Council on the report on Tuesday. He plans to tell the 15-member council that the United Nations will continue to work with Saudi Arabia on the issue and reinforce that only the blacklist is under review, not the substance of the report, a UN diplomatic source said.
In a rare move, Ban - who steps down at the end of the year after a decade in the top UN job - publicly slammed Saudi Arabia for exerting unacceptable pressure on the world body over the children and armed conflict report. Diplomatic sources told Reuters in June that Riyadh suggested a fatwa could be placed on the world body.
"He came out of this bruised and unhappy," said a second UN diplomatic source, speaking on condition of anonymity.
Ban has since met with Saudi Deputy Crown Prince Mohammed bin Salman and Saudi Foreign Minister Adel al-Jubeir. Two weeks ago Saudi Arabia sent a letter that diplomatic sources said did not address UN concerns about the risks to children in Yemen and was described by one source as "superficial."
A second letter received by the United Nations last Thursday "does not address yet all of our concerns, but is good enough to continue with the joint evaluation," said one of the diplomatic sources, speaking on condition of anonymity.
"They are willing to continue to be engaged, they formally accept international humanitarian law, and give all sorts of info useful to avoid and prevent future incidents affecting children," said the source.
However, the same source said the United Nations needs "more specifics," citing as an example a commitment to international humanitarian law as "too general."
"We think we sent a fairly comprehensive letter that I hope has served to address all the concerns of UN and clarify all issues surrounding the report by the UN," Saudi UN Ambassador Abdallah Al-Mouallimi told Reuters on Monday.
UN sanctions monitors said in January that the Saudi-led coalition had targeted civilians in Yemen and that some of the attacks could be crimes against humanity, sparking calls by rights groups for the United States and Britain to halt sales of weapons to Saudi Arabia that could be used in such attacks.