Japan's Osaka battled past Swiss lucky loser Viktorija Golubic 2-6, 7-5, 6-1 after 27-year-old Badosa cast more doubts over her fitness ahead of the French Open as the Spanish former world number two pulled out of another tournament due to a recurring back injury.
Kvitova was forced to withdraw due to a leg problem, so three-time Grand Slam finalist Jabeur advanced via walkover and will next face sixth seed Italian Jasmine Paolini or qualifier Lulu Sun.
Czech Kvitova, 35, secured her first victory since returning from maternity leave by defeating Irina-Camelia Begu in the first round.