Bayindir is thought to be on Erik ten Hag's shortlist of replacements for Dean Henderson, who is in talks over a permanent move to Crystal Palace.
The Turkey international has been Fenerbahce's number one shot-stopper since leaving Ankaragucu in 2019, amassing 44 clean sheets in 145 matches for the club.
Bayindir - who stands at 6ft 6in tall - helped Fenerbahce win the Turkish Cup in the 2022-23 season, while he has earned five caps for his national team and was called up to their Euro 2020 squad.
The 1998-born shot-stopper still has four years left to run on his contract with Fenerbahce, having penned an extension in March, but he could be tempted by a move to Man United, even if he does play second fiddle to Andre Onana.
However, The Athletic states that Man United want to get to the bottom of a back concern that ruled Bayindir out of the final few weeks of the 2022-23 season, and he is currently undergoing early medical tests with the Red Devils.
The 25-year-old has already played four times this term since recovering from the issue - keeping two clean sheets - but he has been benched for Fenerbahce's two most recent games.
Should the provisional checks provide reassuring results, Man United could supposedly close a deal for Bayindir for as little as €5m (£4.3m), assuming that Henderson departs as expected.
The England international will seemingly not be making his loan move to Nottingham Forest permanent any more, as the Tricky Trees poached Matt Turner from Arsenal instead.
However, Palace are hopeful of striking a loan-to-buy deal for Henderson, who could become the second goalkeeper to leave the Red Devils this summer after Bayer Leverkusen-bound Matej Kovar.
Onana and Tom Heaton would represent Ten Hag's only two senior goalkeepers if Henderson leaves, but the latter is in the final year of his contract and has also been struggling with a calf strain.
With Henderson in talks over an exit and Heaton in the treatment room, Ten Hag called up 19-year-old Radek Vitek for last weekend's 2-0 loss at Tottenham Hotspur, but the Dutchman is now searching for a more experienced number two.
In addition to Bayindir, the Red Devils are said to have taken an interest in Benfica and Greece shot-stopper Odysseas Vlachodimos, who allegedly wants to leave the Portuguese champions after a falling-out with head coach Roger Schmidt.
The Red Devils are back in action at home to Nottingham Forest on Saturday afternoon before a daunting trip to the Emirates to meet Arsenal on September 3.