Argentinean Pizzi would represent Saudi Arabia at the World Cup draw in Moscow on Friday, they added.
Pizzi is the third coach to fill the position in the last three months after Bauza was hired to replace Bert van Marwijk, who qualified the country for their first World Cup finals since 2006.
Van Marwijk left the post after failing to agree a new contract with the Saudi federation while Bauza was fired last week after just five games in charge, including back-to-back losses against Bulgaria and Portugal.
Pizzi was most recently in charge of Chile, who he took to the final of this year's Confederations Cup, but the 49-year-old resigned after failing to steer the country to the World Cup finals.
Saudi Arabia have qualified for their fifth World Cup, with the country's best performance coming in their debut appearance at the finals in 1994 in the United States, when they reached the Round of 16.
The country has been eliminated in the group phase of the competition on each occasion since.
(Reporting by Michael Church in Bangkok, Editing by Peter Rutehrford/Amlan Chakraborty)
