The Daily Telegraph claims that Van Dijk was Klopp's priority defensive target for the summer and that he did not pursue an alternative centre-back in hopes of bringing the Netherlands international to Anfield towards the end of the transfer window.
But with the transfer having failed to materialise, the paper says Liverpool will renew their interest in Van Dijk in January, or even wait until next summer.
Liverpool publicly ended their interest in Van Dijk in June after Southampton threatened to report the Reds to the Premier League over an alleged illegal approach for the player.
Reds owners Fenway Sports Group assured Southampton that they would not make any further approach for the player unless they were invited by the south coast club.
"This is someone making a tough decision to leave one great club and come to another great club and I think he had many options, so to get him means I am delighted," Klopp told Liverpool's official website.
"I hear a lot of talk about positions but let's talk about the player – he is someone with really good abilities who always gives everything for his team. A player that is positive and willing to take risks to try and make positive things happen.
"He may still be young but he has a lot of experience in the Premier League, Champions League and for his country."
Oxlade-Chamberlain could make his Liverpool debut against Manchester City at the Etihad Stadium on 9 September.