Yet despite their current struggle for survival, the Swans will not be short of reasons to be hopeful.
As well as winning two of their past four matches at Old Trafford, their boss Paul Clement will be able to call on a particularly personal inspiration.
The 44-year-old was in the away end at the 'Theatre of Dreams' 12 years ago, watching his brother Neil play for West Bromwich Albion in their bid to pull off the most dramatic of escapes from relegation.
The Baggies had been bottom of the Premier League at Christmas but a remarkable turnaround in the second half of the season saw them visit Manchester United on the penultimate weekend with their hopes of survival somehow intact.
"I actually remember going to Old Trafford that season when they got a draw. It was 1-1 and proved to be a massive point," Clement recalls.
"I was sitting up in the corner with the West Brom fans, went up from London on the train. I was a coach at Fulham in the academy at the time.
"I didn't go to the last game. They were bottom going into the last day of the season and played Portsmouth at home. The bottom three were changing - the great escape."
It was indeed a great escape.
West Brom started the final day of the 2004-05 campaign at the foot of the table and, despite leading Portsmouth 2-0, were heading to the Championship with 82 minutes gone as Crystal Palace led against Charlton.
But Jonathan Fortune's equaliser for the Addicks - and defeats for Norwich and Southampton - secured survival for West Brom, whose jubilant fans spilled on to the Hawthorns pitch at full-time to celebrate an escape Harry Houdini might have deemed farfetched.
The similarities to Swansea's current situation are striking.
When Clement was appointed the club's third boss of the season in January, the Swans were bottom of the Premier League.
They won five of their next eight league games to climb out of the relegation zone but a subsequent run of six matches without a win dumped them back into the bottom three.
Last Saturday's win over Stoke lifted morale, yet the joy was tempered by Hull's victory against Watford which kept Swansea two points adrift of safety.
Hull remain Swansea's chief competitors in their battle to avoid the drop and the Tigers are away at Southampton on Saturday, although Clement insists that game will have no bearing on his approach at Old Trafford.
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"No. I've decided that already," he says.
"Whatever happens in that game, we're going to approach United in the same way because we know there's different tests coming after that and that's not going to change.
"I think we just have to focus on our performance and that's really what we did very well when we went to Liverpool [and won in January].
"Forget it's Liverpool, forget it's Anfield, focus on a good performance and we'll see what happens. That's what we're here to do."
January's improbable win over Liverpool breathed new life into Swansea's bid for survival, and victory at Manchester United would have a similarly galvanising effect.
It would be a significant upset too, as United are aiming to set a new club record of 25 league matches unbeaten in a single top-flight season.
However, Swansea can take heart from the fact that no ground has staged more draws in the Premier League this season than the nine at Old Trafford.
"That can be the mentality of the opponent. When you know you're playing someone so strong, you can defend deep, make it difficult for the opposition to break you down," Clement says.
"One thing they haven't done is lost there since Man City, which was right at the start of the season.
"They've had too many draws, haven't they, to push them up the table, and they've also had injury troubles, which is more now than before.
"We have to perform to our best and hope the effects of the games they play and their injuries will hinder them in some way. But it will not be easy."
Facing Manchester United at Old Trafford is a daunting task for most sides, so it would be understandable if Swansea City, deep in relegation strife, travelled there on Sunday with some trepidation.