Jurgen Klopp's Liverpool missed out on the Premier League title to Pep Guardiola's Manchester City last season
The clubs and their fan groups held a meeting in November after which they all committed to improving relations.
The two sides have sent a joint letter to supporters asking them to report any incidents "that cause harm".
The move comes before they meet at Etihad Stadium in the Carabao Cup fourth round on Thursday.
"The passion and enthusiasm that we see from our fans when we play each other - at the Etihad Stadium or Anfield - is part of what makes matches between our clubs so special," said the letter from Liverpool chief executive Billy Hogan and City counterpart Ferran Soriano.
"We know you are committed to creating a stadium environment where everyone is welcomed, accepted and has a fantastic matchday experience.
"Regrettably, based on recent meetings between our clubs, we also know that a minority of fans are responsible for behaviours and actions that have no place in our game.
"We are therefore writing to ask for your full support in tackling these behaviours and to ensure that both of our clubs' good names are preserved."
Liverpool beat Manchester City 1-0 at home in the Premier League in October when they last met.
City boss Pep Guardiola said members of the Anfield crowd threw coins at him during the game, while Liverpool condemned "vile" chanting from the away end.
The Manchester club were also privately unhappy with pre-match comments from Reds boss Jurgen Klopp about their spending power, with the German later denying his remarks were xenophobic, while City also claimed the windscreen of their bus was damaged as it left Anfield.
The recent incidents when the two clubs meet also date back to when the City team coach was attacked prior to a Champions League defeat at Liverpool in April 2018.
"You can help us by reporting any incidents... so they can be investigated," continued the letter.
"We appreciate that some fans may not be aware of the impact of their behaviours on other fans, both inside and outside the stadium, and we will continue to work with our fans on this education.
"It is however important for everyone to be aware that we are fully committed to dealing with unacceptable behaviours and will take action by issuing bans to those found to be responsible.
"If we all stand together to report the behaviours and actions that cause harm - including inappropriate gestures, chanting, throwing of objects, as well as any and all forms of discrimination - then we can all concentrate on supporting our teams."
-- Courtesy of BBC Sport