Barnes played for Dalglish during the Scot's first spell in charge at Anfield.
Barnes, who also worked with Dalglish at Celtic, said: "If they do well enough I think he will stay [in charge].
"But let's get to the end of the season first and see how they do.
"They will improve and I expect Liverpool to be in a better position than where they are now. Then of course there is a decision to make.
"So it really does depend on how well Kenny does between now and the end of the season."