Ranieri had recognised his future at the West London side was uncertain after Wednesday's stinging 2-0 defeat at Southampton.
Arsenal manager Arsene Wenger spoke last week of his desire to keep the 24-year-old Oxlade-Chamberlain, who was also a target for champions Chelsea.
Premier League champions Chelsea eased past a disappointing Everton with goals from Spaniards Cesc Fabregas and Alvaro Morata sealing a 2-0 victory on Sunday.
Liverpool scored three goals in 12 dramatic first-half minutes, two of them from Emre Can, on their way to a hugely entertaining 4-2 win over Hoffenheim which took them back to the Champions League group stage after a two-season absence on Wednesday.
Title-chasing Tottenham Hotspur made hay in the sunshine as Son Heung-min scored twice in a 4-0 thrashing of Watford that closed the gap on Premier League leaders Chelsea to four points in Saturday's early kickoff.
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.
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Borussia Dortmund completed their spectacular implosion this season, losing 2-1 at struggling Werder Bremen.
The challenge against champions Chlesea on Sunday is for Manchester United to show that they can replicate the confident and businesslike win that City showed a month ago.
'The silky touch, the swerving runs, the parabola pass perfection, the goals -- Salah on a football pitch these days is a sporting experience so celestial you feel inebriated just witnessing it.' 'It is football of the gods: Pure, eye-popping and incomprehensible,' applauds Dhruv Munjal.
Photos from the Premier League matches played on Sunday
Manchester City's all-conquering season reached new peaks as Sergio Aguero became the club's all-time top scorer with a decisive goal in a thrilling 4-2 victory at Napoli that put them into the Champions League last 16 on Wednesday.
Ferguson, who makes the comments in the documentary 'Sir Alex Ferguson: Secrets of Success', to be screened on BBC One on October 31, also defends David Moyes' ill-fated appointment as his successor.
Liverpool manager Brendan Rodgers has dismissed speculation about his job by saying he is the best man to lead the club and that he expects to be in charge next season.
Angelo Ogbonna leapt high to power home a header in the dying seconds of extra time to claim a timely first goal for the club as West Ham United sank Liverpool 2-1 on Tuesday to reach the last 16 of the FA Cup.
Brendan Rodgers has admitted that his job is on the line, but has pledged to keep working towards improving Liverpool's results whether he stays on as manager for "one day more, one month, another year, or whatever".
Irresistible Chelsea go into the Christmas round of fixtures knowing they can end the year by re-writing the record books.
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Images from the English Premier League matches played on Sunday
Pogba suffered the injury in United's 3-0 Champions League win over Basel this month.
Barca captain Messi opened the scoring with a "Panenka" penalty chipped straight down the middle in the 17th minute after Luis Suarez was fouled in the area, putting the home side ahead in the tie after a goalless first leg in France.
Photos from the English Premier League matches played on Sunday
Liverpool coach Juergen Klopp had praised Spanish clubs on the eve of the match, saying they had better scouting, coaches and player development than their English, German and Italian equivalents.
Cristiano Ronaldo scored twice in his 150th European appearance to give Real Madrid a 3-1 victory at Borussia Dortmund, their first against them in Germany, and make it two wins out of two matches in their Champions League Group H.
Manchester City made sure of third place in the Premier League when they romped to a 5-0 win at Watford on Sunday to ensure a Champions League berth for next season.
London clubs' erratic start to the Premier League season improved with victories for Chelsea, Tottenham Hotspur and Arsenal on Saturday but Manchester sides continued to make the running, with City and United the top two.
Manchester City set a new top-flight record of 15 consecutive wins as their seemingly unstoppable march towards the English Premier League title continued with a 4-0 romp at bottom club Swansea City.
Chelsea ticked off three more points in their Premier League title quest with a 2-1 victory over Manchester City on Wednesday but Tottenham Hotspur's Houdini-like 3-1 win at Swansea means the race is far from over.
Tottenham Hotspur manager Mauricio Pochettino punched the Anfield turf in frustration at the final whistle after his side drew 1-1 with Liverpool in a Premier League match full of thrilling, attacking action on Saturday.
Liverpool striker Daniel Sturridge was substituted with a thigh injury as the English Premier League club beat Bayern Munich 3-0 in a pre-season match in Germany on Tuesday.
Watford climbed into the top four of the Premier League after Tom Cleverley's 92nd-minute goal gave them a 2-1 victory over Arsenal at Vicarage Road on Saturday.
Chelsea moved 12 points clear of Arsenal, who suffered their fourth defeat in nine league matches, while fourth-placed Liverpool fell 13 point adrift after their wretched 2017 continued with a shock 2-0 defeat at resurgent Hull City.
Borussia Dortmund edged past Freiburg 1-0 on Sunday courtesy of captain Sebastian Kehl's first goal in almost two years to go four points clear in second place in the Bundesliga.
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How quickly affections can switch in soccer. Where once they would serenade coach Jose Mourinho at Stamford Bridge with an operatic flourish, now Chelsea fans have fallen headlong for another man driving them inexorably to the Premier League title.
Bernardo Silva has made an encouraging start to the Premier League campaign and the Manchester City midfielder said that feeling more settled at his second season with the champions was a key factor in his performances.
IMAGES from the English Premier League matches played on Wednesday
Fans are openly speculating that owner Roman Abramovich, a man who places notoriously high demands on his managers, could call time on Jose Mourinho's second spell at Stamford Bridge if Liverpool can inflict another defeat on the champions.
Images from Champions League matches played across Europe on Tuesday
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