The 20-year-old full-back broke his tibia and fibula following a challenge by PSV defender Hector Moreno in the 15th minute in the Red Devils opening Champions League group stage game midweek.
Nani shared a picture of Shaw alongside the caption: “Luke my friend, wish you a great and fast recovery! All the best! #fastrecovery.”
United were beaten 2-1 in the Netherlands by PSV on Tuesday night but the game was overshadowed by the injury to the England international.
Shaw is not expected back until at least March next year according to head coach Louis van Gaal.
“I am not a doctor so I cannot say that, but when you have a double fracture it is I think until six months,” Van Gaal was quoted as saying on Sky Sports.
Nani joined Fenerbahçe on a three-year deal over the summer transfer window and already has three goals and two assists for the Yellow Canaries.
The Portuguese international is expected to start for Fenerbahçe against Bursaspor in the Turkish Super League on the weekend.