Trippier, 28, is on the verge of a £20m move to La Liga side Atletico Madrid.
Rose, 29, has been granted time off “to explore prospective opportunities with other clubs”, along with Vincent Janssen and Georges-Kevin Nkoudou.
The Danish midfielder said in June that he “might want to try something new” following speculation about his future.
Also last month, Rose admitted that he did not know whether he would still be with Spurs next season.
Ben Davies, Eric Dier and Cameron Carter-Vickers have stayed in London for rehabilitation, while Serge Aurier, Davinson Sanchez and Victor Wanyama will return for pre-season training next week.