The Swans went down 2-1 to Blackburn Rovers, while United lost 1-0 against Birmingham City on their return to the second tier.
Russell Martin's first competitive game in charge of Swansea did not go to plan as last season's beaten playoff finalists contributed to their own downfall at Ewood Park.
City did at least get off the mark with an impressive 3-0 win at Reading in round one of the EFL Cup in midweek, but points are needed on the board in the Championship.
The Welsh side have not started a league campaign with successive losses since 2013-14 when in the Premier League, and they have beaten United in each of the last three meetings.
That is something the Blades will of course be eager to change this weekend as they look to respond to their home loss at the hands of Birmingham.
Maxime Colin headed in the only goal of the game at Bramall Lane with 19 minutes played, meaning defeat for United in their first game since dropping out of the top flight.
Like Swansea, the Blades also have a new man in the dugout as Slavisa Jokanovic - who has twice earned promotion with Watford and Fulham - took charge on a three-year deal.
Jokanovic admitted to being disappointed with the result against Birmingham, though he fell short of criticising his players for their performance.
United bounced back with a midweek cup win against Carlisle United, but they have now won just seven of their 39 league games since the start of last season.
United's win against Carlisle was most notable for Rhian Brewster scoring his first goal for the club after a 1,357-minute wait spanning the past 10 months.
Having been used as a substitute on the opening weekend of the season, the former Liverpool man - who impressed on loan at Swansea in the past - may get the nod up top here.
Goalkeeper Aaron Ramsdale has been strongly linked with a move to Arsenal this week, but he is expected to feature between the sticks in South Wales.
As for Swansea, Martin made five changes to his lineup for the Reading win and may be tempted to stick with a similar XI this weekend given the nature of his side's victory.
The Swans this week brought in Flynn Downes from Ipswich Town and the midfielder is in contention for a swift debut.
Swansea City possible starting lineup:
Benda; Cabango, Naughton, Latibeaudiere; Naughton, Paterson, Downes, Grimes, Bidwell; Lowe, Piroe
Sheffield United possible starting lineup:
Ramsdale; Baldock, Basham, Egan, Norrington-Davies; Berge, Norwood, Fleck; McGoldrick, Brewster, Burke
We say: Swansea City 2-1 Sheffield United
Martin and Jokanovic could do with getting off the mark with their first league wins in charge of Swansea and United respectively, but at least one of them will be disappointed.
Swansea were let down by some sloppy moments against Blackburn but looked impressive against Reading, so we are backing the Swans to pick up all three points on home soil.