Liverpool made a late bid to try and sign Caicedo from Brighton at a time when Chelsea were in advanced talks.
But the Ecuador international remained hellbent on signing for the Blues and eventually made his way to Stamford Bridge in a British record deal worth £115million.
It left Liverpool empty-handed but Klopp has seen the funny side of things.
After being asked about new signing Wataru Endo, the Reds coach said: "Firstly, I'm glad we can announce an agreement with the club AND the player."
It comes after Klopp said in his press conference last week that Liverpool and Brighton had struck an agreement over a fee for Caicedo.
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Chelsea also beat the Reds to Romeo Lavia, who was announced as a new signing at Stamford Bridge on Friday morning.
When asked about the double deal, Klopp was in a less than fun mood.
He replied: "I've nothing to say about that, absolutely nothing to say.
"The transfer business is that you talk to people, clubs, agent, players and sometimes it works, sometimes it doesn't. Nothing else to say from my point of view."
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But the arrival of Endo has been a major boost after losing Jordan Henderson, Fabinho, James Milner, Alex Oxlade-Chamberlain and Naby Keita in midfield.
"I'm absolutely happy about having Wataru in our team," Klopp said.
"I'm from Germany, I watch a lot of Bundesliga, I've known him since he signed for Stuttgart and I liked him from the start there.
"He's a bit of a late bloomer, a lot of people said he was already too old when he joined Stuttgart, but I think differently. I always liked him.
"The last week was obviously a tricky one, no doubt about that, but when you have a problem you can either stick with the problem or you find a solution and I'm more than happy to have the solution.
"Comparing them [Endo with Caicedo and Lavia] footblaling wise, they're a similar profile, a similar position.
"He's a really good footballer. He's 5ft 10in but jumps through the ceiling, he's good in the air, defensive challenges he loves it, technically he's really smart in closing down the right gaps and players.
"He's also got a big heart and desire - it's a really good package, so I'm a really happy guy."