The second-seeded Mayer earned a 6-2, 0-6, 6-4 victory over Argentina's Carlos Berlocq, while the fourth-seeded Andujar cruised to a 6-2, 6-3 win over Romanian wild card Marius Copil.
Mayer, the runner-up to Nikolay Davydenko in Munich earlier this year, is still trying for his first career ATP singles title. He'll next face Spain's Albert Ramos, a 2-6, 6-4, 7-5 winner over fellow Spaniard Pere Riba, in Friday's quarterfinals.
Andujar, who claimed his first title earlier this year in Casablanca, is trying to reach the final of this event for the second year in a row. He lost to Juan Ignacio Chela in last year's title match and could face the Argentine this weekend.
Chela is this week's top seed and had already advanced to the quarterfinals with a win on Wednesday. He'll take on Italian seventh seed Andreas Seppi on Friday, with the winner of that match to face either Andujar or Italian qualifier Alessandro Giannessi in the semis.
Italy's Filippo Volandri took out third-seeded Spaniard Marcel Granollers on Thursday with a 6-1, 6-3 rout and will next meet Brazil's Joao Souza, who followed up his first-round upset of Spain's Tommy Robredo with a 4-6, 7-6 (8-6), 6-1 triumph over Romanian wild card Adrian Ungur on Thursday.
The 2011 Bucharest champion will earn $93,000.