A week after losing to the Greek upstart at the same stage of the Madrid Open, the 32-year-old Nadal will be eager to prove that defeat was just a blip as the Spaniard fine-tunes his game ahead of his bid for a 12th Roland Garros title.
The 20-year-old Greek progressed after Roger Federer pulled out with a right leg injury before their quarter-final clash.
Nadal went 2-0 down against Verdasco before surviving three break points at 4-4 in the first set, but the second seed pulled away by winning eight consecutive games to continue his hunt for a ninth title in Rome.
Argentina's Diego Schwartzman reached the last four with a 6-4 6-2 win over Japanese sixth seed Kei Nishikori and will face world number one Novak Djokovic or Juan Martin del Potro.
In the women's draw, Konta booked her first semi-final spot in Rome with a 6-3 3-6 6-1 win over Czech Marketa Vondrousova.
The Briton, who defeated Americans Sloane Stephens and Venus Williams in her two matches on Thursday, will face sixth seed Kiki Bertens on Saturday for a place in the final.
Madrid Open champion Bertens reached the last four after Japan's world number one Naomi Osaka pulled out with a thumb injury ahead of their quarter-final.
Konta, celebrating her 28th birthday, hit 42 winners against the 19-year-old Vondrousova and dominated the final set to clinch victory in one hour 53 minutes.
She heads towards this month's French Open in fine form on clay, having finished runner-up in the Morocco Open at the start of May.
Fourth seed Karolina Pliskova recorded her 500th career win with a 6-7 (5-7) 6-2 6-2 victory over Victoria Azarenka.
She will face either Kristina Mladenovic of France or Greek Maria Sakkari in the other semi-final.
