Smalling and Young were on crutches at the end of last term following operations while Nani has also required surgery during the close season.
The players are due to fly to Bangkok next week for the first leg of the tour with Saturday's encounter against the Singha All-Star XI on the horizon - the manager's first game in charge of the Reds.
"There's a doubt about Ashley Young and Chris Smalling," stated Moyes. "Nani has had an operation on his nose.
"There are a few [doubts] but they may still go on tour. Certainly Chris and Ashley we'll assess at the start of next week."
Wilfried Zaha will link up with his new team-mates on Monday after representing England Under-21s this summer, but Shinji Kagawa and Javier Hernandez are due more time off.
"At the moment, Zaha is due in for training on Monday," revealed the manager. "Kagawa will meet us in Japan and be involved there. Hernandez isn't going to come out because he's just finished with the Confederations Cup. He'll go back to the Aon Training Complex and prepare there."