Van Dijk officially completed his move from Southampton to Merseyside on Monday morning, having trained for the first time at Melwood the previous day.
Klopp has continually stressed the need for patience with his new club-record signing, and speaking after the 2-1 win at Turf Moor, he highlighted his plans for an adjustment period.
“I don’t know [when Van Dijk will make his debut],” he told reporters.
“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.”