Liverpool spent close to £40million on the Brazil worldwide just two days after their Champions League final defeat against Real Madrid and the former Monaco star was expected to slot in at the base of Klopp's midfield.
Klavan is expected to complete a move to Italian Serie A club Cagliari for a fee of €2m and is on his way to undergo a medical in Italy.
The transfer happens just hours after Jurgen Klopp warned European clubs against trying to sign Liverpool players on the cheap after the closure of the transfer window in England.
"We need to stay positive, very good, but stay angry as well". We are not in a financial need situation where we have to do that.
"Some of us have been operating at a very high level for many years, not sure Klopp can say the same".
"I was never in doubt about Joe and his ability to play at centre-half, but he needed time after a long injury", Klopp said of his defensive options at a news conference on Friday.More news: Serena Williams dominates in return at Cincinnati Masters
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"I'm hearing about it for the first time now, so it doesn't seem like it's close", he said.
With Dejan Lovren late back from the World Cup and Joel Matip injured, the 21-year-old was given the chance to play for the first team and managed to stake his claim in excellent fashion. Can we give them the opportunity if he wants to leave?
"But I really think we could give a few players the opportunity - but we have to see how many".
'Atletico have always had an ambitious project, we can get anywhere with work, humility. It's getting better and better but we are still in a process.