Barnes, who scored 28 goals in the 1989/90 season, the Reds' last title win, believes King Kenny can lead the club to success once again with a couple of additions to the squad once the transfer window opens.
"The timescale (of challenging for the league) depends on the quality of the players they can get in," said the former Red.
"If they don't bring much more quality in to this squad, then I don't think they can win the title next year, they can only progress.
"But with a little bit more investment, getting two or three quality players into the squad, I think they could challenge next year. But it really does depend on who they get.
"Since Kenny has been there, they have had a fantastic run. And I feel that they can challenge for a Champions League place next year.
"I still don't think the current squad is good enough to win the league, but Kenny has maximised the potential of his squad, he is making them play consistently well, and they will definitely be challenging for honours - or at least for a place back in the Champions League - next season.
"The pressure will be on, especially having had the run that they have had since Kenny has been there. People are naturally expecting them to continue into next season and to be right up there, which I believe they can be.
"However, after the euphoria of Kenny being back, and the last half of the season going so well, we have to be realistic in what we expect for next year. That is why, with Kenny maximising the potential of this squad, the potential is there to challenge for a Champions League place.
"To say that this squad can actually win the league? No. I still think it needs to be strengthened in certain areas."