Klopp said he was making a pragmatic decision to ease the burden on his first-choice players after five matches in 15 days, with a busy festive programme also coming up
Edwards has been a key figure in Liverpool's successful recruitment programme, but the 42-year-old is ending his 10-year association as he wants a fresh challenge.
Jurgen Klopp's Liverpool side were one game away from setting a new club record of 26 games unbeaten in all competitions but three costly errors from goalkeeper Alisson Becker cost them dear.
It was just the sixth time Liverpool failed to win a Premier League home game after leading by two goals.
The Athletic said "sources close to the Liverpool bench" thought Manchester-born Taylor did not send off more United players because he "took pity" on them.
City's record run in the competition came to an end at the London Stadium as the Hammers emerged 5-3 winners from a penalty shootout after both sides were guilty of missing plenty of chances in a 0-0 draw.
It was Liverpool's biggest win against United since 1925 and moved them seven points clear of Solskjaer's side in the Premier League table.
Premier League: Liverpool manager Jurgen Klopp still wary of struggling Manchester United's 'quality'
Liverpool's gifted front three of Mohamed Salah, Sadio Mane and Roberto Firmino are all scoring freely for the unbeaten Reds, who are just one point behind leaders Chelsea.
Having moved to the top of the group with victories over AC Milan and Porto, the six-time European Cup winners head to Madrid to face the Spanish champions Atletico.
Premier League: Jurgen Klopp says Mohamed Salah 'best in the world' after Liverpool's lopsided win at Watford
For the first time in his career, Salah has scored in eight successive top-flight matches and Liverpool boss Klopp has no doubts he is now better than even Lionel Messi and Cristiano Ronaldo.
Klopp expects Newcastle to fulfil their ambitions eventually, but he suggested it might be a rocky road to the top because money alone cannot guarantee success.
Liverpool's habit of avoiding giving big contracts to players aged over 30 has fuelled fears Salah will leave Anfield.
Only seven of the 20 Premier League clubs have more than half of their squads fully vaccinated, according to the most recent data.
Salah's current deal is set to expire in 2023 and the 29-year-old knows his worth to Klopp's men as negotiations over a new deal have dragged on.
Take-up of the vaccine has been mixed among elite athletes, with reports last week suggesting only seven of the 20 Premier League clubs have over 50 percent of their squads fully vaccinated.
European football matchday: Unbeaten Liverpool face tough test against Man City; Fiorentina host Napoli
Liverpool's unbeaten start to the Premier League will be put to the test with the visit of defending champion Manchester City to Anfield
The two head coaches have won the last four Premier League trophies, with City claiming three, and a fierce rivalry has developed between the clubs.
Liverpool have scored 20 times in 17 days and have 26 goals in their first nine fixtures to equal a club record for a top-flight campaign set in 1922
Champions League: Liverpool need to show a reaction in Europe after Brentford mishaps, says Jurgen Klopp
Jurgen Klopp played down fears over Liverpool's defensive frailties after their 3-3 draw with Brentford