The Dutch worldwide was injured after a challenge from Everton goalkeeper Jordan Pickford during the early stages of Saturday's Merseyside Derby and substituted minutes later.
Klopp said that Van Dijk has preferred to spend time alone to come to terms with the nature of his injury, but stressed that both he and the club will be there "like a good wife" to support the Dutchman.
It is a nightmare scenario for Liverpool. "They lost (Dejan) Lovren before the start of the season. others in those positions are injury prone".
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The 23-year-old Nat Phillips, with just one FA Cup appearance to his name either side of two loan spells at Stuttgart last season, is now the back-up option with the highly-rated but unproven 17-year-old Billy Koumetio another alternative.
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On the reaction to the Spurs defeat, Solskjaer said: "It's one of those things that you have to get over as soon as you can and move on".
Liverpool did not give a timescale on Van Dijk's absence, but an injury of that nature will likely keep the Netherlands global out for a long period.
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'Virgil played for us I don't know how many games in a row.
"Joe Gomez played a sensational game".
"I didn't see it back, it took a while so obviously the offside was not too big and Pickford kicks him completely", he told Sky Sports.
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Dominic Calvert-Lewin's towering header restored parity with nine minutes left but Everton were then reduced to 10 men when Richarlison was sent off for a wild challenge on Thiago.
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Sections of the Dutch media have shared their anger at Jordan Pickford not picking up a red card for the challenge, and there's upset ay Van Dijk's club and country being deprived of the player's services for the foreseeable future.
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