Zirkzee was recently confirmed as the first signing of the INEOS era, with the Dutchman arriving from Bologna, but his involvement at Euro 2024 means that he is yet to link up with his new teammates.
There is also no Tyrell Malacia, with Man United revealing that the left-back, who missed the entirety of last season with a serious knee injury, is still training individually.
"Tyrell Malacia is absent from the squad as he continues his rehabilitation from injury under the guidance of medical staff at Carrington," read a statement on Man United's official website.
The final eight first-team players absent from the tour squad were involved with their countries over the summer and will therefore not link back up with the squad until the start of August.
Fernandes, Martinez are among the absentees
Altay Bayindir, Diogo Dalot, Bruno Fernandes, Kobbie Mainoo and Luke Shaw all represented their countries at Euro 2024 and are therefore on extended breaks.
Meanwhile, Lisandro Martinez, Alejandro Garnacho and Facundo Pellistri featured at the Copa America and are also having rest periods before returning to club duties.
Three players involved in Euro 2024 are in the tour squad, though, with Denmark duo Rasmus Hojlund Christian Eriksen included alongside Scotland's Scott McTominay.
Tom Heaton, who was in the England goalkeeping group at the European Championship, has also been named in the squad, having penned a new one-year contract this summer.
Is new signing Yoro in the Man United squad?
New signing Leny Yoro has been included in the tour squad, as the 18-year-old did not represent France at Euro 2024 and is also not in action for his country at the Olympics.
The centre-back made his Red Devils debut against Rangers on Saturday afternoon, playing 45 minutes of his side's 2-0 success in Edinburgh.
There are also spots in the squad for Victor Lindelof, Aaron Wan-Bissaka and Jadon Sancho despite the transfer speculation surrounding the trio.
Man United will open their tour against Arsenal on July 27 and will then take on Real Betis and Liverpool in quick succession before heading back to England for the Community Shield against Manchester City.