Konate, 26, has been heavily linked with a move to the La Liga giants with his contract at Liverpool set to expire at the end of the season.
The Frenchman came under fire for some disappointing performances earlier in the season but appears to have turned things around over the past couple of months.
Speaking after their 1-0 Premier League win over Sunderland on Wednesday, Van Dijk urged Konate to commit his future to the club.
"We are friends, we speak about everything," Van Dijk told reporters.
"It's a process and let's see what comes out of it. It's never that easy. We saw with my own situation last year so it's never that easy that we can just say 'let's get it done'.
"Obviously I want him to stay," Van Dijk added.
"He's an important figure on the pitch. That's what everyone sees but off the pitch as well, he's one of the leaders. He's outstanding and in my eyes, a world-class centre back.
“I can do only so much but it's in the club's hands, together with his agents and himself so let's see what comes out of it but I have no influence otherwise on that."