Asked if a deal could be reached by Sunday night, Azevedo told Reuters: "I hope so."
He spoke as he was entering what could turn out to be one of the last negotiating sessions in the global trade body's 12 year pursuit of a trade deal, capping 10 weeks of round the clock talks that Azevedo has overseen since he took the helm of the WTO on September 1.
The deal would streamline customs procedures worldwide, making border-crossing processes more predictable and transparent. Studies by the World Bank and the Organisation for Economic Cooperation and Development have said it would add hundreds of billions of dollars to the world economy.
It also includes some changes to the rules on agriculture and some special treatment for the poorest countries.
Azevedo forced the marathon negotiations to try to get the deal before a meeting of WTO ministers in Bali in the first week of December.
Diplomats say that if the 159 WTO members can agree on the text at a meeting of the WTO's General Council in Geneva, ministers would rubber-stamp it in Bali, with little chance of negotiations being reopened.
A deal in Bali would revive confidence at the WTO after its credibility ebbed away over the past decade with the slow death of the Doha round of talks, which gave way to the more modest package of reforms now under discussion.
A meeting of the General Council was due on Thursday, but it was postponed while negotiations on the trade reform package continued. Azevedo said he was not yet sure if he would be able to call a meeting of the council this weekend.
(Reporting by Tom Miles; Editing by Alison Williams)