Monaco, who won the event in 2007 and is seeded No. 2 this year, defeated local favorite Andreas Haider-Maurer 6-3, 6-4 while piling up six breaks. Monaco won 56 percent of the points when receiving serve.
Verdasco, the third seed and event runner-up in 2005, took out Guillermo Garcia-Lopez 6-2, 7-6 (9-7). He also had six breaks on the day.
Dominic Thiem pulled off a 7-5, 6-3 upset of fourth-seeded Jurgen Melzer in an all-Austrian match. Thiem took nearly three-quarters of the points on serve and lost serve just once.
The quarterfinals are set, with Monaco meeting Gimeno-Traver and Thiem playing Montanes in the lower half of the bracket. Haase goes against Verdasco and Leonardo Mayer faces eighth-seeded Marcel Granollers.
Caroline Berg Eriksen: Soccer player's wife triggers debate with post-birth selfie
Exploding whale video goes viral on Internet