Klopp made the surprise announcement on Friday that he will be vacating his post as Reds manager when the 2023-24 campaign comes to a close, while coaching trio Pep Lijnders, Peter Krawietz and Vitor Matos will also leave the club at the same time.
Van Dijk was signed by Klopp in January 2017 from Southampton for a then club-record £75m and he has since established himself as one of the world's most imposing central defenders.
The Dutchman scored his 21st goals for Liverpool in his 246th appearance for the club when they beat Norwich City 5-2 at Anfield in the FA Cup fourth round on Sunday.
Van Dijk was an integral part of the Liverpool teams that won the 2018-19 Champions League, the 2019-20 Premier League title, as well as the FA Cup and EFL Cup in 2021-22.
Van Dijk was appointed as Liverpool captain last summer following the departure of Jordan Henderson, but the defender has since cast doubt over his future at Anfield following Klopp's announcement.
Asked if he sees himself part of the post-Klopp era at Liverpool, Van Dijk said, as quoted by Sky Sports News: "That's a big question. I don't know."
"The club will have a big job on their hands [after Klopp], that is well known. To replace not only the manager but the whole staff and there are so many things that will change.
"I'm very curious which direction that will go in but when that will be announced we will see our situation.
"It will be the end of Jurgen Klopp's era - I am still part of it that's why I don't like to talk about it - and that is my main focus.
"Hopefully we will have the success we all dream of and by then probably there will be more clarification about what the club wants for the future and then we will see."
Van Dijk will turn 33 years of age in July and will also enter the final year of his Liverpool contract this summer.
The uncertain future of Van Dijk comes at a time when star attacker Mohamed Salah continues to be linked with a big-money move to Saudi Arabia.
Liverpool currently sit five points clear at the top of the Premier League table and will lock horns with Chelsea at Anfield in their next match on Wednesday night.