The 31-year-old landed awkwardly on his left shoulder after reacting to a blocked shot from Sergio Reguilon, and he immediately waved to the dugout to receive treatment.
Pope was eventually replaced by Martin Dubravka in the 86th minute and gingerly walked down the tunnel whilst supported by two members of Newcastle's medical team.
The Magpies held on for all three points against the Red Devils, but Eddie Howe expressed his concern for Pope after the match, stating that he could be without his number-one goalkeeper for a "lengthy period".
"That's the disappointment on the night," Howe told reporters as quoted by talkSPORT. "Nick looks like he's dislocated his shoulder. Such a strange thing, really, because he'd have made that dive thousands of times.
"It looks like maybe the arm has just kept on going, maybe on the moist floor, and his shoulder has come out of his joint. We're going to have to seek specialist opinion and then see what happens, but it doesn't look good.
"Too early to say (a timeframe). He was in pain when the shoulder was out of his joint. It's gone back in now and he feels better but we'll seek a specialist opinion. It doesn't sound good.
"It's a massive blow for us if Nick's going to be out for a lengthy period of time," Howe added. "It looks really innocuous, they look the most serious ones at times. I think he's dislocated his shoulder just diving, the arm kept going and looks like we could lose him."
According to talkSPORT, Pope is set to see a specialist on Monday to determine the severity of his shoulder injury, but he is expected to be ruled out for up to four months.
A separate report from The Telegraph claims that Pope could be sidelined for between four and five months following an initial assessment after the win over Man United.
Pope has been an integral player for the Magpies since joining the club in the summer of 2022, keeping 26 clean sheets in 62 appearances across all competitions, including eight in 20 games this season.
The former Burney shot-stopper sustained a similar shoulder injury during his time with the Clarets in a Europa League qualifier in July 2018 and was subsequently ruled out for five months.
Pope's lengthy absence from the Newcastle side could therefore hinder his hopes of earning a place in Gareth Southgate's England squad for Euro 2024.
The goalkeeper is also the latest Newcastle player to be added to the club's growing injury list that includes Jacob Murphy (shoulder), Sven Botman (knee), Joe Willock (Achilles), Callum Wilson, Matt Targett (both hamstring), Dan Burn, Elliot Anderson (both back), Sean Longstaff (ankle), Harvey Barnes (foot) and Javi Manquillo (groin).
Dubravka is expected to feature regular as Newcastle's goalkeeper in Pope's absence, while Loris Karius and Mark Gillespie are other backup options at Howe's disposal.
Newcastle have reportedly identified former Man United goalkeeper David de Gea, who is currently a free agent, as a potential target.