Mr. Pedersen was speaking following a meeting with the government and opposition co-chairs of the Syrian Constitutional Committee, who have agreed to start the process for constitutional reform.
The members of its so-called "small body", tasked with preparing and drafting the Constitution, are in the Swiss city for their sixth round of talks in two years, which begin on Monday.
Their last meeting, held in January, ended without progress, and the UN envoy has been negotiating between the parties on a way forward.
“The two Co-Chairs now agree that we will not only prepare for constitutional reform, but we will prepare and start drafting for constitutional reform," Mr. Pedersen journalists.
"So, the new thing this week is that we will actually be starting a drafting process for constitutional reform in Syria."
The UN continues to support efforts towards a Syrian-owned and led political solution to end more than a decade of war that has killed upwards of 350,000 people and left 13 million in need of humanitarian aid.