Sebian winger Markovic managed to escape Anfield last month after a tortuous four-and-a-half year spell that saw him make just 19 Premier League appearances.
The 24-year-old joined the Reds back in the summer of 2014 for £20m, making all 34 of his appearances for the club in the 2014/15 season.
He his since spent the rest of his Liverpool career with clubs across Europe, spending time on loan with Fenerbache, Sporting Lisbon, Hull and Anderlecht.
However, Markovic was unable to force another move last summer, loan or permanent, leaving him with the U23s after Klopp confirmed there was no place for him in his title chasing squad.
Two Premier League 2 matches and a goal later, he has been given a second chance in English football by relegation battlers Fulham. And speaking on Friday, Klopp revealed how pleased he was the former Benfica man had found a new club.
The German also opened up on how difficult it had been for both himself and the player to see his career stall at the club.
“For him, it’s brilliant. I can imagine it was unbelievably difficult for him”, Klopp told LFC TV.
“It was not too easy for me – that’s not too important – having a player like him in the wider squad and not using him. But it was clear, Lazar knew for a long time how our plans are, and for different reasons transfers didn’t work out.
“We were not responsible for that. Now he can go to Fulham and use and show his potential again, which I’m happy for.
“He’s a good lad and a really good footballer – hopefully it will work out for him.”
Markovic could make his debut for Fulham today against Crystal Palace, but is unlikely to due to a lack of match fitness. Liverpool, meanwhile, travel to West Ham on Monday night as they look to build on their lead at the top of the table.