Egyptian diplomat Moushira Khattab took third place, with 12 votes, in the second round of voting to elect the new UNESCO director-general, Al-Ahram Arabic news website reported.
Qatar's former culture minister Hamad Bin Abdulaziz Al-Kawari took the lead with 20 votes, followed by France's candidate Audrey Azoulay, who received 13 votes.
Fifty-eight members of UNESCO's executive board voted in a first round on Monday to select a successor to outgoing Bulgarian Director-General Irina Bokova. Khattab also finishing third.
Delegates will continue to vote in further ballots this week if no candidate obtains the required 30 votes.
A third and fourth round of voting is set for Wednesday and Thursday.
A fifth and final round is scheduled for Friday between the two candidates who obtain the most votes in the fourth round.
In case of a tie between the two finalists, a toss will determine the winner.
The winner is expected to be announced no later than 13 October.
Six candidates from France, China, Vietnam, Egypt, Qatar and Lebanon are in the running for the leadership of the organisation, after three of the initial contenders withdrew.
In UNESCO's 72-year history, no Arab nation has ever held the position of director-general.
Egypt's foreign ministry spokesman said in comments to state TV that Qatar's first-round win in voting for a new UNESCO director-general raises "many question marks," after the Gulf nation's candidate took a surprising lead following Monday's secret ballot.
The Egypt-Qatar rivalry in the UNESCO vote comes amid an ongoing diplomatic row between Qatar and a number of Arab nations, including Egypt. The group of four nations has imposed a boycott on Qatar, accusing it of sponsoring terrorism, which Doha has denied.
Abu Zeid said that the first-round success for Qatar is no indication of the final outcome.