The German is expected to leave Anfield once his current contract expires in 2024.
On Klopp's ideal replacement, Barnes told BettingOdds.com: "Pep Guardiola. Of course there are other managers that you would think are better suited, such as Steven Gerrard who has had a very good season at Rangers, but is he going to come to Liverpool off the back of one good year? Of course he would be given time because of his relationship with Liverpool.
"Probably not because you would want a manager with more experience and is proven at the highest level. He would be given time because he's Steven Gerrard, but would that be the right thing for Liverpool? Maybe not.
"I'm sure Steven will do really well over the course of his career, but he's going to have to show a longer level of consistency over a longer period of time before we start linking him with the Liverpool job."