France's Audrey Azoulay, a former culture minister like al-Kawari, was in second place among the six candidates still in the running to replace outgoing UNESCO director-general Irina Bokova.
Al-Kawari garnered 20 votes from the 58 board members gathered in Paris since last Friday, according to results posted on Twitter by board chairman Michael Worbs of Germany on Tuesday.
Thirty votes are needed to clinch the nomination to head the Paris-based UN Educational, Scientific and Cultural Organization.
The candidate selected by the board must be approved by UNESCO's 195 member states in November, but this is seen as a formality.
There were 13 votes for Azoulay in the secret ballot, Worbs tweeted.
In third place was Egyptian career diplomat Moushira Khattab with 12 votes.
China's Tang Qian and Vietnam's Pham Sanh Chau each scored five votes, while Vera El Khoury of Lebanon won three.
If no candidate wins an outright majority after four rounds of voting, it goes to a run-off between the top two.
Most of the candidates acknowledge the need to reform the 71-year-old organisation whose bloated bureaucracy is accused of inefficiency.
UNESCO has been accused of bias in the Israeli-Palestinian conflict, and it infuriated Israel and staunch ally the United States by granting full membership to Palestine in 2011.
Both countries suspended their funding to the agency -- best-known for its prestigious World Heritage List -- over the move.
Arab countries have complained that UNESCO has never had a boss from their region.
However, UNESCO does not observe the kind of rotation by world region which is used when choosing a UN secretary general.
Three candidates have dropped out of contention -- those from Guatemala and Iraq before the board began its deliberations -- and that of Azerbaijan after Monday's first round.