Van Dijk became the world’s most expensive defender after the Reds struck a £75m deal with the Saints last month, with the centre-back making the official move to Anfield on Monday after the opening of the January transfer window.
He will now face competition from Ragnar Klavan, Dejan Lovren and Joel Matip for a starting spot on Merseyside but Klopp insists he does not plan to throw the defender straight into the mix.
“I don’t know [when Van Dijk will make his debut],” said the boss.
“We made a few tests that make sense only for him and for us, to see where he stands. That’s all. There is no rush.
“He is on fire and he really wants it, that’s all good, but we have to make sure he can perform. Centre-half is a position which depends on all the others on the pitch. There are a few challenges you have to solve by yourself, all the rest is always together. That means you have to tune it so it works really together.
“We need time. A pre-season is for that usually but we don’t have a pre-season. Everything will be fine.
“After the Everton game we have a little bit more time, that will help, but it’s a long-term project – so it’s not important when he makes his first game, it’s important how many good games he will play for Liverpool. That’s what I’m interested in.”