Wednesday's report, which found that Liverpool fans had been entirely blameless for the crush at the 1989 FA Cup semi-final that left 96 of them dead, was widely welcomed, although it remains to be seen whether their deaths will stop the club being mocked with hateful songs by rival fans.
A minority of Manchester United fans still sing about Hillsborough while some Liverpool supporters continue to gloat over the 1958 Munich air crash which decimated the United team.
"In terms of the chants, I speak as a human being and I never like to hear anything like that, whatever clubs it is, that associates people and other people's tragedies and death," Rodgers told Liverpool's website.
"Unfortunately you have a very, very small percentage of idiots at every club that will always try and smear a club's reputation. So, of course, it's obvious that these are certainly chants that no-one wants to hear about any club."
"You would hope that maybe this is a line in the sand in terms of how the supporters behave with one another," Ferguson said on Friday.
"We are two great clubs and we should understand each other's problems in the past. Certainly the reputation of both clubs doesn't deserve it."
It is hoped that the fall-out from the Hillsborough report, which disclosed a police cover-up and serious failings by the authorities on the day of the Britain's worst football disaster, will lead to more goodwill among fans.
Saturday's West London derby between Queens Park Rangers and Chelsea, however, will put that to the test.
It will be the first time that the clubs have met since former England captain John Terry was cleared of racially abusing QPR's Anton Ferdinand in the corresponding fixture last season.
The situation is further complicated because Terry's team mate Ashley Cole, who is expected to play after recovering from injury, was a character witness for the defence in July.
Chelsea manager Roberto Di Matteo said his players would "respect" the handshake while his opposite number Mark Hughes said they would be governed by the Premier League.
QPR's website issued a reminder about behaviour on Thursday while Chelsea also said on their website that "abuse and discrimination" have no place in a football stadium.
