Having allowed Dean Henderson to join Nottingham Forest on loan for the 2022-23 campaign, the Red Devils only have Tom Heaton as senior back-up to David de Gea this season, with Lee Grant retiring over the summer.
Man United boss Erik ten Hag is said to be determined to bring a new stopper to the club before the deadline, and according to Italian journalist Fabrizio Romano, Vlachodimos has emerged as a target.
Romano claims that the 28-year-old is 'one of four names' included on the English club's shortlist.
However, Ajax are also believed to be interested in the stopper, and it will not be a straightforward deal due to Benfica's desire to keep him at the club for the 2022-23 season.
Vlachodimos, who has been capped on 27 occasions by Greece, started his professional career at Stuttgart before moving to Panathinaikos in 2016.
The goalkeeper spent two years with the Greek outfit before arriving at Benfica, and he has made 174 appearances for the Portuguese giants during his time at the club.
Vlachodimos has already featured on five occasions at club level this term, keeping three clean sheets in the process, including in league outings against Arouca and Casa Pia.
Man United are believed to be looking to sign a back-up for De Gea, but the Spaniard's performance against Brentford last weekend was heavily criticised, and it is understood that Ten Hag wants a player capable of putting severe pressure on the current number one at Old Trafford.
The Red Devils also have young goalkeepers Matej Kovar and Nathan Bishop at the club, but both are expected to leave on loan before the end of the transfer window.
According to the Manchester Evening News, Kovar and Bishop have agreed to join Cheltenham Town and Wycombe Wanderers respectively but remain at the club, with Man United using four goalkeepers in first-team training.
Kovar made the trip with the first-team squad to Brentford last weekend and will allegedly be allowed to depart if a new goalkeeper arrives in the near future.
Bishop spent last season on loan at Mansfield Town, meanwhile, and the 22-year-old is also set to move on before the end of the window.
Watford's Daniel Bachmann and Newcastle United's Karl Darlow are also among the stoppers to be linked with moves to Ten Hag's side during the current market.