1 seed Hubert Hurkacz of Poland in the quarterfinals of the European Open on Friday in Antwerp, Belgium.
He'll meet Frenchman Richard Gasquet in the semis; Gasquet beat Belgium's David Goffin 6-2, 7-6 (6).
De Minaur outlasted Canadian fourth seed Denis Shapovalov 6-2, 7-6 (4) and Rune turned the tables on second seed Cameron Norrie of Great Britain 6-7 (4), 6-3, 6-3.
Musetti will be joined in the semis by countryman and second seed Matteo Berrettini, who beat Japan's Taro Daniel 6-2, 6-3.
Berrettini's opponent will be Mackenzie McDonald, an American who rallied past Chinese qualifier Zhizhen Zhang 4-6, 6-3, 6-2.