Besiktas supporters jeered on-loan Karius as they crashed out of the Turkish Cup to second-tier side Büyükşehir Belediye Erzurumspor on Wednesday.
The German has conceded 38 goals in 24 appearances in all competitions and reports now claim he has told his agent to find him a new club after being turned on by fans, reports the Liverpool Echo.
Karius is more than halfway through a two-year loan deal with the Turkish outfit, who paid Liverpool an initial fee of £2.25million.
Besiktas are obligated to buy him for a further £7.25million in the summer of 2020 if certain performance criteria are met related to appearances and European qualification.
His future at Besiktas looked uncertain last season after he was heavily criticised by then-manager Senol Gunes and had to ask FIFA to take action over unpaid wages.