But now the Reds are reigning Champions League winners and are on the verge of clinching their first Premier League title in 30 years.
And Liverpool legend Barnes says United have allowed their standards to slip, just as his team did at Anfield in the 90's.
"In the way when I first went to Liverpool, they were a successful team before I was there and the demands to win and the pressure to win was the most important thing, regardless of whether they had superstars like Ian Rush, Kenny Dalglish and they had won everything.
"The demand to win everything was there. Then, all of a sudden, Manchester United came along in the early nineties and things had changed.
"If you look at the way the players were after say 1992-93, Liverpool still was a huge club, and it seemed that the players there were happy to be at Liverpool without necessarily recognising the responsibility to win.
"So when we were finishing third in the league, fourth in the league, we were still Liverpool players, bigger than 99 per cent of the teams, even if they finished above us, Arsenal whoever, but we were the Liverpool players without the demands to win.
"So we still got that kudos, that feeling that we are great, that we are better than everyone else because we are Liverpool players.
"Now with Manchester United, when Ryan [Giggs] was there and he came through and they started to win all the Premier League trophies, regardless of whether they won the Premier League the year before, they had that demand that they put on themselves to win.
"Then I think after that Man City took over, Manchester United still were the biggest club, and the players were happy to be with Manchester United without recognising the responsibility of having to win.