The Red Devils are said to have identified the Greece international as an ideal backup to Andre Onana amid speculation surrounding the long-term future of Dean Henderson.
Crystal Palace are said to be in talks over a deal for Henderson, while Matej Kovar has already departed Old Trafford to join German giants Bayer Leverkusen.
As a result, Tom Heaton is the only senior backup to Onana at present, but the 37-year-old is currently nursing a calf injury and has less than a year left to run on his contract.
With Heaton sidelined, 19-year-old Radek Vitek was named on the bench for the weekend's 2-0 defeat to Tottenham Hotspur, but Erik ten Hag is understandably searching for a more experienced deputy.
According to the Daily Mail, Vlachodimos is now looking for a way out of Benfica following a tiff with head coach Roger Schmidt, who did not include him in the squad for Saturday's Primeira Liga win over Estrela.
During the opening weekend of the season, Vlachodimos had shipped three goals in a 3-2 loss to Boavista, after which Schmidt was critical of the 29-year-old's performance.
The Benfica boss subsequently made the bold decision to drop Vlachodimos in favour of 21-year-old Samuel Soares, while new arrival Anatoliy Trubin warmed the bench, and the risk paid off as the Eagles won 2-0.
Following his falling-out with Schmidt, Vlachodimos believes that an exit from Benfica would be best for all parties, but he only signed a new deal until 2027 back in March.
However, given his age and his desire to find a new club before the transfer window closes, Man United are unlikely to be quoted a hefty sum of money to sign Vlachodimos in the coming weeks.
The former Stuttgart and Panathinaikos man joined Benfica for a mere £2.1m in 2018, and he has 94 clean sheets to show from 225 matches for the Portuguese champions in all tournaments.
Vlachodimos - who was born in Germany and stands at 6ft 3in tall - kept 21 clean sheets in 34 games en route to his second Primeira Liga title last season, while he has also won two Portuguese Super Cups with the Eagles.
A 33-cap Greece international, Vlachodimos previously represented Germany up to Under-21 level and won the 2017 Under-21 European Championships with his birth nation.
The 29-year-old could become Man United's fourth arrival of the window after Mason Mount, Onana and Rasmus Hojlund, but the Red Devils are still said to be keen on signing another new defender and midfielder before September 1.
Ten Hag's side will look to recover from their defeat to Tottenham when they host Nottingham Forest on Saturday afternoon.