The Red Devils signed Andre Onana from Inter Milan last month, and the Cameroonian will start the 2023-24 campaign as the club's number one, replacing David de Gea, who has departed on a free transfer.
Despite Onana's arrival, there is still huge uncertainty in the goalkeeping department at the club, with Dean Henderson being heavily linked with a move to Nottingham Forest, while there is also speculation surrounding the future of the veteran Tom Heaton.
As it stands, Heaton is number two to Onana, with Man United head coach Erik ten Hag said to be reluctant to lose him, but the 37-year-old has attracted attention from the likes of Everton and Luton Town.
Nathan Bishop has now been sold to Sunderland, while Matej Kovar and Radek Vitek are also expected to be on the move before the end of the transfer window.
As a result, the 20-time English champions are in the market for another goalkeeper, with Urawa Red Diamonds's Zion Suzuki being linked with a move to the club.
However, there has also been speculation surrounding Fenerbahce's Bayindir, and according to CNN Turk, Man United have now launched a move for the goalkeeper.
The report claims that an official attempt to bring in the five-time Turkey international has been made, with the stopper having a €6m (£5.2m) release clause in his current deal.
There have previously been suggestions that Bayindir was not prepared to make the move to Old Trafford unless he was given a proper opportunity to become their number one ahead of Onana.
Bayindir started his professional career at Ankaragucu in 2016, spending three years in the first team at the club ahead of a switch to Fenerbahce.
The stopper has made 143 appearances for his current side in all competitions, keeping 44 clean sheets in the process, including two outings during the 2023-24 campaign.
Bayindir played the full 90 minutes in both legs of Fenerbahce's Europa Conference League second qualification round contest against Zimbru Chisinau, which the Turkish outfit won 9-0 on aggregate.
Man United are currently preparing for their final two friendlies of the summer, facing Lens and Athletic Bilbao on Saturday and Sunday respectively, before beginning their 2023-24 Premier League campaign at home to Wolverhampton Wanderers on August 14.