The 25-year-old was responsible for two of Real Madrid's goals in last season's 3-1 Champions League final defeat, leading to widespread ridicule.
Karius broke down on the field in Kiev and later apologised for his high-profile blunders, while many of his fellow pros came to his defence.
However, the Daily Mail claims that the support began to waver when the former Mainz 05 stopper featured in a professionally-produced Instagram video six weeks later, seemingly showing him without a care in the world.
The video was later taken down, but the report indicates that the damage had already been done and any trust between the keeper and his teammates had eroded, leading to his summer exit.
Karius has joined Besiktas on an initial two-year loan deal that will eventually become a permanent switch.