The Chinese candidate Tang Qian, who serves as UNESCO's Assistant-Director General for Education, and Lebanese global law expert Vera El Khoury respectively had five and four votes only.
"Now more than ever UNESCO needs a project... which restores confidence and overcomes political divisions", the foreign ministry said in a statement reacting to the U.S. pullout.
Khattab acquired 13 votes in the third round, and with the Chinese candidate's withdrawal she has the chance to garner at least 18 votes in the upcoming.
Qatar's Hamad bin Abdulaziz Al-Kawari and France's Audrey Azoulay both received 18 votes each in the third round on Wednesday to replace UNESCO's director-general Irina Bokova by the start of 2018.
Egypt submitted an official note to UNESCO's outgoing chief, Irina Bokova, on Friday morning to request the verification of "violations that have been detected throughout the electoral process", foreign ministry spokesman Ahmed Abu Zeid said in a statement.
The election of UNESCO's new chief has been narrowed down to two candidates, one from Qatar and the other from France.
The rift continued to fester in recent years, with the organisation becoming the scene of repeated diplomatic flare-ups after efforts led by Arab countries to pass resolutions critical of Israel.
Egyptian Foreign Minister Sameh Shoukry, who was in Paris for the vote, "urged all his friends to vote for France" instead of Qatar in the final round, a member of Khattab's delegation told AFP.
A fifth round will be held in case no candidate gets the absolute majority, but it is not required to get 30 votes this time.
UNESCO's general assembly will have to sign off on the board's pick.