The 29-year-old is expected to leave the Portuguese champions in the coming days after an alleged falling out with manager Roger Schmidt.
Vlachodimos's performances were crucial in Benfica's title triumph last season, as he kept 21 clean sheets in 34 Primeira Liga games, but he shipped three goals on the opening day of the 2023-24 season to Boavista.
After Schmidt publicly criticised Vlachodimos's display, the Greece international was left out of the squad for last weekend's 2-0 win over Estrela, where Samuel Soares began the game and new arrival Anatoliy Trubin warmed the bench.
Vlachodimos extended his Benfica contract until 2027 only five months ago, but an exit is now expected to go through before the end of next week, and Man United have been put on red alert.
According to journalist Florian Plettenberg, the goalkeeper has already said yes to Man United's contract proposal until 2028, and the Red Devils will now negotiate a fee with Benfica.
Despite Vlachodimos's long-running deal with the Eagles, a fee of €15m (£12.9m) should be accepted by Benfica, and Erik ten Hag believes that the Greece international would provide suitable competition to Andre Onana.
However, the Red Devils are yet to make a final decision in their pursuit of a new goalkeeper, as talks over a deal for Fenerbahce shot-stopper Altay Bayindir are also rumbling on in the background.
Bayindir is understood to have undergone pre-medical tests, as Man United wanted reassurances over the Turkey international's recent back injury before tabling an official bid.
A deal for Bayindir could supposedly be done for just €5m (£4.3m) - representing a far cheaper option than Vlachodimos - and Fenerbahce are already believed to have identified a replacement in the shape of Dinamo Zagreb's Dominik Livakovic.
Nevertheless, a transfer for either goalkeeper hinges on the future of current number two Dean Henderson, who is in discussions over a loan-to-buy switch to Crystal Palace after Nottingham Forest instead poached Matt Turner from Arsenal.
All parties are keen for Henderson's move to go through, but a recent report claimed that Palace are yet to make a breakthrough in negotiations with the Red Devils, who have already let Matej Kovar join Bayer Leverkusen this summer.
The German-born Vlachodimos joined Benfica from Panathinaikos for just £2.1m in 2018, and he has since represented the Eagles on 225 occasions, keeping 94 clean sheets in that time.
The 33-cap Greece international has won five major trophies with Benfica, as well as helping the Portuguese champions reach the Champions League quarter-finals in each of the last two seasons.
Man United, who have already signed Onana, Rasmus Hojlund and Mason Mount this summer, seek an immediate return to winning ways in the Premier League against Nottingham Forest on Saturday afternoon.