Anfield supporters last celebrated winning a trophy way back in 2012 when they beat Cardiff in the League Cup final.
But Barnes told the Daily Star: “Well, would we rather get silverware and win the League Cup and finish 25 points behind City or finish second behind City by three points and not win silverware.
“That for me is more important. That shows progress.
“That shows that you're going to be consistently challenging at the highest level for the league and that will mean they'll be challenging in cup competitions.
“So the whole idea of needing silverware, for me, is not as important as getting much closer to the top.
“Because if you're much closer to the top of the Premier League, that means that you're consistently winning matches and playing well which would help you in cup competitions as well.
“Yes it would be nice to win a trophy, but I'd rather not win a trophy and finish second, seven or eight points off the top."