The pair worked together when Klopp managed Borussia Dortmund, but the German revealed that he did not want to sign the midfielder with his latest club Liverpool.
"I had that player. If you have had players, you know more about them. You want to learn about other players. That's how it is," he told the 'Liverpool Echo'.
"I knew quite early that he was going this way [to United]. There was absolutely no reason for me to jump in there. That isn't how we work.
"We didn't look for players we had already worked with. If you do that, you know their good, their bad and you wonder whether you will do it again.
"It's not boring but you know everything about each other so how can you develop? Where is the next step? Where is the next push for development?"
Unlike many of his Premier League counterparts, the Liverpool manager has opted for a number of more low-key signings rather than spending big money on just a few individuals.
Do you think Klopp missed a trick by not trying to sign Mkhitaryan?