Bayindir has joined on a contract running until 2027, with the option of a further year, United said.
The 25-year-old, who has five caps, played 40 times in all competitions for Fenerbahce last season as they won the Turkish Cup and finished second in the Super Lig.
“It is a huge honour to join Manchester United and become the first Turkish player to represent this incredible club,” Bayindir said.
Bayindir has made 166 career appearances and kept 53 clean sheets, according to United’s website. He arrives at Old Trafford a day after goalkeeper Dean Henderson moved to Crystal Palace.
United play Arsenal in the league on Sunday.
($1 = 0.7886 pounds)
(Reporting by Hritika Sharma in Hyderabad; Editing by Peter Rutherford)