Born in Jerusalem in 1953, Savir studied International Relations at the Hebrew University of Jerusalem, where he gained a BA. He went on to work as an administrator, and served general manager of the Ministry of Foreign Affairs. Between 1993 and 1996 he was the Chief Negotiator of the Oslo Accords. His experience became the basis for his book The Process: 1,100 Days that Changed the Middle East, published in 1998.
In the 1999 elections he was elected to the Knesset on the Centre Party's list. On 6 March 2001 he and two other MKs left the Centre Party to form the New Way faction. On 28 March he resigned from the Knesset, and was replaced by David Magen, who represented the Centre Party. During his time in the Knesset he served on a number of committees, including the Foreign Affairs and Defense Committee.