The Netherlands international was on the receiving end of a heavy challenge from Toffees goalkeeper Jordan Pickford and had to be subbed off after only 11 minutes.
England’s Pickford was surprisingly not penalised for the challenge and Everton came from behind twice to secure a point against the defending champions.
Asked about Van Dijk’s injury, Klopp said after the 2-2 draw: ‘It’s not good. Virgil plays with pain but he couldn’t play on.
‘Not good. He played so many games in a row but he couldn’t play on.’
Former England and Arsenal defender Alex Scott, meanwhile, says Pickford should ‘absolutely’ have been sent off for his ‘reckless’ challenge on Van Dijk.
She told BBC Radio 5 Live: ‘Jordan Pickford’s challenge on Virgil van Dijk should absolutely have been a red card.
‘It was reckless, the sort of challenge that could end a player’s career. He has to be sent off.
‘It was rash – the prime example of what everyone has been saying about Pickford.’
Ex-Leeds United striker Jermaine Beckford, meanwhile, added: ‘The way Jordan Pickford came out to address Virgil van Dijk and try to tackle him and win the ball off him was really, really dangerous.
‘But, again, it’s kind of what we have become accustomed to seeing from Pickford.’
Sadio Mane gave Liverpool the lead after just three minutes at Goodison Park before Michael Keane equalised for the hosts on 19 minutes.
Mohamed Salah put Klopp’s side ahead in the second half but England’s Dominic Calvert-Lewin restored parity with a header in the 81st minute.
Reds captain Jordan Henderson thought he had scored an injury-time winner but the goal was ruled out by VAR after Mane was adjudged to be in an offside position.
On Liverpool’s performance, Klopp said: ‘The performance was top, it was a top game and a good game for both teams.
‘We were dominant against a side full of confidence and with a clear plan. It was probably the best away game since I was at Liverpool.
‘I saw a really good performance. I would be more happy if we win but I wanted to see a response and see the team we were before.’