Leicester are third in the table after nine games -- eight points adrift of leaders Liverpool and two behind City. The two teams above them are the only sides to win more points since Rodgers was appointed in March.
Manchester City paid the price for missed chances as they were held to a 1-1 draw at Burnley on Saturday with Johann Berg Gudmundsson's 82nd minute equaliser earning the hosts a share of the points.
Southampton's rich vein of form in the Premier League continued as they beat Burnley 3-1 with two goals from Charlie Austin on Sunday while another promoted side, Middlesbrough, fared just as poorly, losing 1-0 at home to Watford.
A summary of Saturday's matches in the FA Cup.
Burnley struck either side of halftime with headers from Jeff Hendrick and Michael Keane to claim a 2-0 Premier League home win over Watford on Monday in the first top-flight meeting of the two clubs.
Images from the EPL matches played on Saturday
Premier League teams Burnley and Middlesbrough were knocked out of the English Cup by lower-league teams in the second round on Wednesday.
Liverpool were sent spinning to a 2-0 defeat at promoted Burnley in the English Premier League on Saturday while Chelsea came from behind to win 2-1 at Watford thanks to late goals from Michy Batshuayi and Diego Costa.
News of all that's transpired on and off the football field
IMAGES from the English Premier League matches played on Wednesday.
News of all that's transpired on and off the football field.
Liverpool's Sadio Mane marked his return from injury with a late goal to break Norwich City's resistance and secure a 1-0 win on Saturday to extend his side's Premier League lead to 25 points.
News of all that transpired on and off the football field.
Eight wins and a draw in their last nine games had seen Leicester emerge as the closest pursuers of runaway leaders Liverpool but they went down 3-1 at the Etihad despite taking an early lead through Jamie Vardy. Images from the English Premier League matches played on Saturday.
Chelsea's American forward Christian Pulisic scored his first Premier League goals for the club with a hat-trick and Willian added another in an entertaining 4-2 win at Burnley.
News of all that transpired on and off the football field.
Liverpool head to United with widest gulf in years
News of all that transpired on and off the football field
The duo, who escaped unhurt after an attempted carjacking last month, were not in the squad at Newcastle United last weekend due to "further security incidents" which the club did not elaborate on.
Five talking points from the Premier League weekend:
Images from the English Premier League matches played on Tuesday.
Liverpool have been told by a tribunal to pay Championship (second-tier) leaders Burnley up to eight million pounds ($11.7 million) for striker Danny Ings.
The first few months of this season have been disappointing, as Spurs have won only three of their opening 12 Premier League games and are in 14th spot. They were also thrashed 7-2 by Bayern Munich in the Champions League and knocked out of the League Cup by fourth-tier Colchester United.
High drama ensued during the English Premier League matches on Wednesday as Spurs manager Jose Mourinho was booked while Brighton stunned Chelsea via Alireza Jahanbakhsh's superb equaliser to deny the Blues victory. Here is all the EPL action in IMAGES from Wednesday's matches.
The 38-year-old former Tottenham Hotspur, Liverpool and StokeCity striker spent last season at Burnley but his contract ended last month.
Tottenham Hotspur's hopes of moving into the top four stalled in a 2-2 draw at bottom club Norwich City, while several VAR interventions overshadowed the action. Images from EPL matches played on Saturday.
A round-up of Saturday's matches in the English Premier League.
A round-up of Sunday night's matches in the Premier League.
News of all that's transpired on and off the football field.
'Pure, unadulterated football joy has been in short supply for United fans for a long time now,' complains Dhruv Munjal.
News of all that's transpired on and off the football field.
The two previous meetings at Anfield ended in 4-0 and 3-1 defeats for Arsene Wenger's Arsenal teams, but there is a fresh optimism around the club after a successful transfer window.
Manchester United were held 1-1 at Wolverhampton Wanderers and missed out on taking top spot in the English Premier League.
Liverpool took a giant step towards their first title since 1990 on Saturday when their 3-0 win at Bournemouth combined with Manchester United's 2-1 victory at Manchester City to leave the champions 14 points adrift of the Premier League leaders.
Holders and Premier League leaders Arsenal will be at home to Championship (second tier) Burnley in the fourth round of the FA Cup after most of the big guns avoided top flight opponents in Monday's draw.
With United unable to sign a replacement, the pressure will be on 21-year-old England forward Marcus Rashford to deliver the goals in a more central role than he has previously operated in.
Middlesbrough are back in the Premier League after a 1-1 draw with 10-man Brighton & Hove Albion in their multi-million pound promotion decider on Saturday while Burnley sealed the Championship title with a 3-0 victory at Charlton Athletic.
Images from the English Premier League matches played on Monday
Manchester City manager Pep Guardiola hailed his team
City have 89 points, one more than rivals Liverpool, with both teams having three games to play -- and none of them against top six opponents.