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The Wales international, who scored a stunning back-heel effort in Arsenal's 5-1 win at Fulham on Sunday, has scored 59 goals in 338 appearances for Arsenal since joining from Cardiff in 2008.
Manchester United assistant coach Michael Carrick hails Jose Mourinho as managerial 'great' alongside Sir Alex Ferguson
Carrick told Britain's Press Association that the most striking thing about the Portuguese boss was his track record in winning trophies over many years.
Barkley set up Eden Hazard's opening goal before adding his side's second in the 3-0 victory at Southampton on Sunday that briefly moved Chelsea to the top of the Premier League table.
Arsenal have now won six consecutive league matches within a single campaign for the first time since 2016.
Senior figures at United told Press Association Sport that Mourinho would retain his job for the immediate future.
City were last victorious at Anfield in 2003 and have lost 12 of their last 17 visits.
Chilean boss Pellegrini, celebrating his 65th birthday on Sunday, had spent £100 million ($129 million) bringing nine new signings to the London Stadium in the close season.
Hazard struck twice in seven minutes just before half-time to cancel out Sol Bamba's early shock for the hosts at Stamford Bridge before completing his hat-trick from the penalty spot.
Chelsea refused to sell the forward and Hazard, now in his seventh season at Stamford Bridge, is enjoying Sarri's style of play after the more conservative tactics of former Blues bosses.
Karius played 33 times for Liverpool last season but was heavily criticised for making two crucial errors which led to goals in the 3-1 Champions League final defeat to Real Madrid.
Now in his third season at Old Trafford, United might have already seen the best Mourinho has to offer.
Tottenham defied convention and decided to stick with what they had -- prompting criticism from even the club's own fans.
The Merseyside club will look to get back to winning ways against Huddersfield on Saturday, having settled for a draw after conceding two late goals at Bournemouth last weekend.
Mourinho was in particularly dour and uncooperative mood ahead of his team's Premier League meeting with Mauricio Pochettino's Tottenham on Monday.
It appeared Wolves might inflict a first league defeat on the Premier League champions since City's derby defeat to Manchester United in April.