Lingard, 20, was part of United's 2011 Youth Cup triumph and is yet to make an appearance in the first team.
Januzaj, 18, was named the club's Denzil Haroun Young Player of the Year for last season and was included in the matchday squad for Sir Alex Ferguson's final match in charge at West Brom, but failed to make his debut.
Michael Keane, another member of the Class of 2011, has also travelled as cover for Nemanja Vidić after an impressive loan spell at Leicester City last season.
Vidić, Chris Smalling, Ashley Young, Nani, Antonio Valencia, Darren Fletcher and Nick Powell are out injured while David de Gea, Robin van Persie and Shinji Kagawa will link up with the squad during the tour.
Captain Vidić's exclusion is a surprise after he joined Moyes for Friday's pre-tour press conference.
The Serbian has experienced a number of injury concerns over the past two years, playing only 33 times in that period.
Javier Hernández, who like Kagawa played at the Confederations Cup, is also rested.
Wilfried Zaha - Ferguson's last signing - is the only winger included in the squad and the returning Fabio da Silva will offer his twin Rafael competition at right-back.
United's trio of wingers - Young, Nani and Valencia - all underperformed last season and their absences give Zaha a tremendous opportunity to immediately assert himself at United.