It is thought United will pay Fenerbahce £4.3million for the Turkey international, who will provide cover for Andre Onana.
The 25-year-old has played five times for Turkey and his arrival allowed Dean Henderson to complete his move to Crystal Palace.
“It is a huge honour to join Manchester United and become the first Turkish player to represent this incredible club,” Bayindir said.
“I have a passion for success, and I will give everything to help this special group of players to achieve our ambitions.
“I cannot wait to start working with such an experienced goalkeeping unit. We will support each other and drive high standards every day so that each of us is ready to perform whenever called upon.”
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