Suarez sparked controversy ahead of Liverpool's 2-1 defeat at Manchester United, appearing to refuse the hand of Evra - the man who accused him of racial abuse earlier this season. As a result, United boss Sir Alex Ferguson labelled Suarez a "disgrace to Liverpool Football Club".
However, Barnes believes it would have been fake for Suarez to have shaken Evra's hand, and he is convinced that the media is chief culprit for inciting bad blood between the two clubs. In Barnes' eyes, United and Liverpool would have moved past the situation long ago had the media not persisted in keeping it in the public eye.
"I can tell you they won't be making a big issue over either of the Suarez or John Terry situations in other countries," Barnes told ESPN. "The reputation of English football around the world is not going to be damaged at all.
"I don't care if somebody shakes my hand or not. It's not a big deal, it's insincere, it doesn't mean much, they're shaking hands for show. I don't believe they should be put in that situation.
"We're getting carried away with a bunch of incidents here that, in the bigger picture, don't mean much. Football clubs have to react to it because of what we do in the media. The reality is if Suarez scores 50 goals and United win the league, it's not really important.
"We're making a mountain out of a molehill. We are not the custodians of moral value in the world, we think we are but we're not. There's worse things happening in the world, worse things happening in the country, everything should not be laid at footballers' doors."
Barnes is convinced the handshake ritual should have been scrapped at Old Trafford, but he also believes Evra was as much to blame over the next 90 minutes as Suarez. The United man celebrated directly in the face of Suarez at the full-time whistle, and Barnes insists neither club has come out of the day with any credit.
"I assumed when it was decided that there was going to be the ritual handshake that everybody had agreed that they would shake hands," Barnes said. "When I saw what happened, it made me wonder if Suarez had changed his mind at the last minute.
"I can't imagine that this wasn't discussed by Liverpool. If Luis Suarez said, 'I'm not shaking his hand', which he has every right to if he doesn't want to, then Liverpool should have made all attempts not to have this situation whereby they have to come into confrontation with each other.
"It's a big shock, and obviously it got compounded by Patrice Evra at the end and Sir Alex Ferguson at the end. So it's not a good day from a PR perspective for either club."
Kevin Keegan, also formerly of Liverpool, felt Ferguson was wrong to comment so strongly on Suarez at the full-time whistle. Ferguson stated his belief that the striker should never play for the Reds again, something Keegan argued.
"It's a very strong statement and it's unusual to comment on another club's player," Keegan said. "To come out and say a player shouldn't play for another club is wrong.
"Instead of calming this down they've allowed it to escalate. These are the two biggest clubs in English football. Both clubs have handled it badly.
"Today was a chance to put it all behind them. It was a chance to say to the player, 'Shake his hand and get on with it'. But Evra after the game, why would he want to do that? You've won the game; sometimes as a manager you just don't understand players."
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