Strengthened by new attacking signings Diego Costa and Cesc Fabregas, Chelsea delivered the best display of the opening weekend in beating Burnley 3-1 away and meets another promoted side, Leicester, in its second match Saturday.
City was less eye-catching in its 2-0 win at Newcastle but the result served notice that the defending champions won't be giving up its title easily. Next up for City is Liverpool - the team it pipped to the championship in May - at home on Monday.
Things aren't looking so promising for United and its new manager Louis van Gaal, who are looking to avoid a second straight defeat in an away match at Sunderland.
Here are five things to know about the upcoming Premier League matches:
SCHOLES FEARS FOR UNITED
Perhaps the most damning assessment of United following its 2-1 opening-day loss to Swansea came from its former midfield maestro, Paul Scholes.
''I am scared for United,'' Scholes wrote in Thursday's Independent newspaper. ''Genuinely scared that they could go into the wilderness in the same way that Liverpool did in the 1990s.''
After finishing seventh last season in the turbulent reign of David Moyes, there was renewed hope at Old Trafford following the hiring of the experienced Louis van Gaal and an unbeaten record in preseason.
But the home loss to Swansea punctured that optimism, highlighting the deficiencies that still exist in the team - a lack of pace and dynamism in its attack and a lack of experience in defense.
The signing on Wednesday of left-sided defender Marcos Rojo from Sporting Lisbon will help but there's a general feeling that a superstar needs to arrive at United before the summer transfer window closes if the team is to finish in the top four.
Sunderland started with a 2-2 draw at West Bromwich Albion.
City and Liverpool meet at Etihad Stadium in the first blockbuster match of the new season, featuring the teams who occupied the top two positions in May.
Both teams started off this season with victories, although Liverpool's 2-1 win over Southampton was less convincing and showed the reliance the Reds will have on Daniel Sturridge in the absence of Luis Suarez.
Sturridge scored the late winner at Anfield for his 36th goal in his first 50 league appearances for Liverpool - the best record of any of the club's list of great strikers in the Premier League era.
With only Rickie Lambert as a genuine backup for Sturridge, manager Brendan Rodgers is looking to add another forward in the transfer window and Liverpool has been linked strongly with a move for Italy striker Mario Balotelli.
A deal for Balotelli would need to be wrapped up before Friday lunchtime in Britain for his first match for Liverpool to be against his former club.
With Costa already in the goals, Fabregas slotting in nicely in midfield and the defense looking as strong as ever, there doesn't appear to be a weakness in Chelsea's team as it goes for a first championship since 2010.
Most of the plaudits after the Burnley win went to Fabregas, who added incisiveness to a Chelsea forward line that let down manager Jose Mourinho too often last season.
''I think a player like Cesc, every team wants him,'' said Chelsea forward Andre Schuerrle, who scored from one of Fabregas' astute passes.
Except for maybe Arsenal, which turned down the option of signing Fabregas this summer.
Leicester started life back in the Premier League with a 2-2 draw against Everton and can expect a long afternoon of defending at Stamford Bridge on Saturday.
EUROPEAN DISTRACTION FOR ARSENAL
Tied at 0-0 with Besiktas after the first leg of its Champions League playoff, Arsenal's players may have one eye on Wednesday's return game when they play Everton on Saturday.
And that could be dangerous for manager Arsene Wenger, whose team was blown away 3-0 in the corresponding fixture at Goodison Park last season.
That loss sparked a run of five straight victories by Arsenal that saw Wenger's side claim the fourth and final Champions League spot ahead of Everton.
Arsenal's trio of German World Cup winners, Mesut Ozil, Per Mertesacker and Lukas Podolski, could be in line for recalls after missing the 2-1 win over Crystal Palace on Saturday because they lacked match fitness.
PALACE IN TURMOIL
Crystal Palace's amazing escape from relegation last season is a distant memory now after a wretched start to the new campaign that has seen the club lose not just its opening game and but its manager and sporting director.
Tony Pulis, who led Palace to safety from a perilous position, stepped down as manager last week after a disagreement with co-chairman Steve Parish.
And Iain Moody resigned as sporting director on Thursday after being embroiled in an investigation by the English Football Association into inappropriate text messages allegedly sent and received by Moody and then manager Malky Mackay during their time at Cardiff.
Palace hosts West Ham on Saturday, when it's also: Aston Villa vs. Newcastle, Southampton vs. West Brom; and Swansea vs. Burnley.
On Sunday, it's: Tottenham vs. Queens Park Rangers; and Hull vs. Stoke.