Fourth-seed Ferrer will face off against Argentinian eighth seed Juan Martín del Potro for a spot in the tournament's semi-finals.
Wimbledon is the only Grand Slam tournament in which the man from Alicante hasn't reached that milestone.
"It's an extra motivation," Ferrer — who will officially hold the world number three ranking as of Monday — was quoted as saying in Spanish daily 20minutos.
World number 54 Verdasco, meanwhile, is set for a tough duel with home favourite and world number two, the Scot Andy Murray.
Armed with a new trainer in Ricardo Sánchez, the Madrid local will be hoping to revive a career which has seen something of a slump since 2009 when he reached his highest world ranking of seventh place.
The matches will be played 2pm onwards.
Spain's other male tennis ace Rafa Nadal was knocked out in the first round of the 2013 Wimbledon tournament by rank outsider and world number 135 Steve Darcis of Belgium.
The country's top female player, 18th ranked Carla Suarez Navarro, saw her exit in the fourth round at the hands of Czech world number eight Petra Kvitova.