Basel have confirmed the move and Sio is set to play in their UEFA Champions League play-off, first-leg tie against Bulgarian champions Ludogorets Razgrad on Wednesday if the paperwork is completed in time.
The 24-year-old Sio, who signed for Wolfsburg in January 2012, made only ten Bundesliga appearances, including the final ten minutes of Die Wölfe's (Wolves) 2-0 defeat at Hannover 96 last weekend.
In his 18-month spell at the club he was twice sent out on loan at Germans Augsburg then France's Sochaux.
Sio made the last of his three appearances for Côte d'Ivoire when he came off the bench in their 4-1 defeat to Mexico in New Jersey last Wednesday, and already has Swiss league experience having spent three years at FC Sion.
"The move is a great chance for Giovanni and he can establish himself as a first-choice option at FC Basel," Wolfsburg's team manager Klaus Allofs said. "We wish him all the very best for this new challenge."